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ENERGY 



WORK, HEAT AND 
TRANSFORMA TIONS 






Sidney a/'reeve, m.e. 



• » » 

« • • 
* • • 



NEW YORK 

McGRAW-HILL BOOK COMPANY 
1909 



Copyright. 1909, by the McGRAW-HILL BOOK COMPANY 



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©CI.A251.982 



PREFACE 

The earlier chapters of this work are self-explanatory. The 
later ones justify some discussion of point of view. 

The writer is not a physicist. Educated and trained as an 
engineer, his call to the teacher's chair led him to arrange his 
views as to natural principles with greater care than is common 
with engineers. The ideas promulgated in the following pages 
are in answer to questions received from bright-minded students 
a dozen years ago — questions which were sensibly asked, but 
which "stumped" the teacher for years for an adequate, equally 
sensible reply. 

His efforts at the comprehension of thermodynamic action 
have led him to trespass, perhaps, upon the domain of the 
physicist. For the discussion of matters so intricate as molecular 
dynamics a thorough familiarity with experimental and mathe- 
matical physics might seem indispensable. No one can regret 
more than the writer his lack of this. His apologies for the conse- 
quent shortcomings of this little book are profuse in proportion. 

Yet, should this situation arouse criticism or doubt, the 
answer is easy. Why have not the professional physicists long 
ago done this same thing, that it might have been done far 
better? The materials, opportunity and demand have long 
existed. Whatever question may arise as to the significance, or 
even the definition, of the more recent data of experimental 
physics, there can be none as to the long established principles 
of celestial mechanics. There is as little as to the only less 
venerable data as to thermal processes. For nearly a century 
there has been virtually no question as to the mechanical nature 
of heat. Yet these things cannot be accepted by any teacher of 
thermodynamics without enforcing conclusions substantially dif- 
ferent from those now commonly taught. Therefore the writer 
cannot regard the concepts set forth in this little book as aught 
else than the indispensable premises for, rather than the remote 
conclusions from, experimental and mathematical physics. 

The writer would never wilfully question the doctrine that 
accurate data are essential to progress. Scientific concepts cannot 



advance without them. But what has been forgotten is that they 
are a means to an end, not an end in themselves ; and that end is 
the better understanding of nature's ways. Science serves 
humanity only as it substitutes scientific concepts for supersti- 
tious guess-work. The accumulation of unending columns of 
figures and physical constants, even with religious accuracy, is as 
futile for the uplift of the race as is the accumulation of vast 
hoards of dollars, however conscientiously accounted to the last 
penny. Each may be turned to humanitarian ends. But until 
it is, like a prostrate ladder, it constitutes a trap for the feet of 
the unwary, rather than a pathway erected to higher things. 

The book, therefore, is merely an attempt to fit together 
(i) the Newtonian mechanics, (2) the doctrine that heat is 
mode of motion, and (3) the dozen or so well known facts of 
thermal action, into a consistent whole which may serve as an 
engineer's idea of heat and heat-action. It was originally pre- 
pared for publication in the periodical press, and some of the 
earlier portions appeared, in preliminary form, in the columns 
of The Engineer (London). Some traces of this genesis may 
be noticed in the pages of the book. 

The writer wishes to acknowledge his indebtedness to the 
Sheffield Scientific School, of Yale University, for helpful 
facilities for work. 

New Haven, Conn., July, 1909. 



(^ 



Chapter I. 
Chapter IL 
Chapter III. 
Chapter IV. 
Chapter V. 
Chapter VI. 
Chapter VII. 
Chapter VIII. 
Chapter IX. 

Chapter X. 

Chapter XI. 

Chapter XII. 

Chapter XIII. 
Chapter XIV. 
Chapter XV. 
Chapter XVI. 



TABLE OF CONTENTS 

Page 

Mechanical Energy 7 

Free and Vibratory Energies 18 

The Mean Energetic Condition and the Energy-fund 32 

The Two Factors or Dimensions of Energy 48 

The Extreme or Critical Energetic Conditions 61 

The General Nature of Mechanical Energy 78 

What is Heat? 89 

The Thermal Diagram 95 

Mechanical Concepts of Thermal Phenomena .... 106 

A. Pressure and Volume. 

Mechanical Concepts of Pressure and Volume 

(cont.) 117 

The Two Basic Thermal Processes : Heat-transfer 

and Work-performance 129 

Mechanical Concepts of Thermal Phenomena .... 141 

B. Temperature and Entropy. 

The Energetic Cycle 158 

Reversed and Irregular Cycles 175 

Thermal Equilibrium 182 

Transformations and Conservations 208 



ENERGY 

CHAPTER I. 

Mechanical Energy. 

The muscular system of our modern body politic is its array 
of energy-producing machines. Man has magnified his own tiny 
energies with power borrowed from nature. His land-carriers 
put to scorn the elephant ; his ships make pygmies of the whales. 
Take from mankind to-morrow this artificial multiplication of 
its abilities to "overcome resistance," and within a month its 
ranks will be decimated by starvation. Within a decade it will 
have become a mob of howling beasts, fighting for the insufficient 
means of existence — slipped back centuries in growth of civiliza- 
tion as well as of population. 

Yet to-day, in spite of this overwhelming importance of 
energy in modern community-life, there exists no idea of its 
nature more exact than the general one that it is an aid in the 
overcoming of resistance. Force times space, or mass under- 
going acceleration, equals energy. Thus far we appear to pro- 
ceed coherently. But it is not far enough, when the chief 
business of an increasing fraction of the human race is the 
transformation and transportation of energy — mechanical, hy- 
draulic, electrical, chemical, thermal, etc. ; and even in going that 
far we quite lack, it appears, a complete agreement between the 
authorities. 

Then, again, the bulk of all this current supply of energy to 
the human race is obtained in the form of heat. Yet as to what 
heat is we know virtually nothing. It is commonly described, in 
an attempt to explain its nature, as a "mode of motion" be- 
tween the particles of the hot body. But an explanation of an 
obscure thing, in order to help rather than hinder the under- 
standing, must speak in terms of familiar things ; and when this 
"explanation" of heat in terms of mass and motion is once 
examined, it is found to contain two elements which are even 
more unfamiliar and obscure than is heat itself. 

7 



8 ENERGY 

The first of these elements is that "perfect elasticity" which 
must be attributed to the constantly rebounding particles upon 
the molecular theory of heat. For, since heat continues indefi- 
nitely in a hot body, so long as none is abstracted, each rebound 
must occur millions of times per second, with perfect elasticity, 
in order that the continuous existence of heat may be explained 
mechanically. 

Yet such a thing as a rebound, after collision, with perfect 
elasticity, is unknown in nature. In fact, it is one of the 
fundamental doctrines of thermal science that motion cannot 
occur in nature without some loss of energy inelastically, in 
either friction or impact. If there be in nature such a thing as 
perfect elasticity, human observation has never yet discerned it. 
Therefore, the "mode of motion" explanation of heat is an 
explanation of one thing, namely, heat, which, albeit mystical 
in its nature, is yet familiar to every schoolboy, in terms of 
another thing, namely, perfect elasticity, which no one has ever 
yet known to exist anywhere. The only thing plain about such 
an explanation is that it doesn't explain. 

The other element of common sense which is wanting in this 
alleged explanation of heat is an exact and familiar idea of 
mechanical energy itself. It is of no use to explain heat as a 
microscopic form of mechanical energy if we do not know what 
mechanical energy is. The term mechanical energy, it is true, is 
used most familiarly by engineers ; yet, astounding as the fact 
may seem, it is nevertheless true that there exists to-day no 
agreement between the authorities as to just what mechanical 
energy is. The statements concerning it which are rife among 
the engineers, the teachers of engineering and their text-books, 
can all be reduced quickly to either absurdities or approxima- 
tions. To "explain," therefore, that heat is a "mode of motion," 
when not one student of engineering in a thousand has ever 
been taught what modes of motion are possible in nature and 
what are impossible, is also a procedure of doubtful utility. 

Nevertheless, there is no escape from the overwhelming and 
rapidly accumulating evidence that heat is capable of a real and 
true explanation, as an intricate form of mechanical energy. 
But progress can be made in this idea only by a careful in- 
vestigation, first, of what mechanical energy itself truly is, in 
nature; and, secondly, of what conclusions should be pointed 



MECHANICAL ENERGY 9 

therefrom as to the nature of heat, in natural common sense. 
Fortunately, every student of engineering science is equipped 
for both of these tasks, although few may have been led to 
perform them. A knowledge of Kepler's and Newton's laws of 
motion and force, a little analytical geometry, and enough of the 
concepts of the calculus to permit thinking of millions of mole- 
cules at once, without becoming confused as to what they may 
and may not do — this is all that is required. 

Mechanical Energy. Mechanical force — that is to say, 
force manifested in such a way that the human understanding 
can follow its origin, dimensions and destination — is found in 
nature, exerted through space, in only two elementary ways. 
The first is when gravitation, which acts at all times between 
each two portions of matter in the universe to hold them 
together, finds chance to move them and to destroy their relative 
separation. Such action is called a manifestation of ''potential" 
or "positional" or "space" energy, visible most familiarly in the 
falling of weights. The other instance occurs when either of the 
forces which everywhere and at all times tend to hold each two 
bodies in the universe both together and apart finds chance to 
produce or destroy their relative motion. Such action is called a 
manifestation of "kinetic" or "accelerative" or "motion" energy. 
It is visible in pure mechanics only in the action of centrifugal 
force. 

It is to be noted carefully — much more carefully than the 
text-books require — that it is neither space nor motion alone 
which .constitutes energy, but change in space or motion. A 
suspended weight possesses no energy if it never can fall. A 
flying cannon-ball can overcome no resistance if nothing ever 
interferes with it to stop it. 

But there is another fact which is of even greater importance 
in these definitions, and which the text-books quite omit alto- 
gether. This is that in any energetic manifestation there are 
always involved at least two bodies. No single body may ever 
possess energy. When it is remembered that all of the com- 
monly taught mathematical expressions for energy include but a 
single symbol for mass, and that in general terms energy is uni- 
versally described as an attribute of mass, this statement cannot 
be too strongly emphasized. Indeed, the only proper way to 
state this idea is to say that energy can exist only where mass 



10 ENERGY 

is not — namely, between two mass-portions. It is just as broad 
a truth to say that mas? and energy cannot occupy the same 
place at the same time as it is to say that two portions of matter 
cannot do so. 

To explain more in detail : A ton of water in a mill-pond, 
we say, possesses energy. That is, it will overcome resistance in 
its fall toward the tail-race- But why will it fall, and whither? 
What gives it its weight and energy ? 

Its separation from the earth gives it its energy. So soon as 
it succeeds in reuniting itself with the earth its energy is gone. 
The energy cannot be said to belong to the water, for without 
the propinquity of the earth the water would have no energy. It 
cannot be said to belong to the earth, for all the earth's gigantic 
mass would be inert and helpless to perform work were it a truly 
solid unit, with no fragments split off, like the water, to help it 
embody energy. And, finally, the energy cannot be said even to 
belong to the earth and water together, because both earth and 
water might be present — as is the case, indeed, in the great 
oceans — without embodying any potential hydraulic energy, be- 
cause there exists no separation of earth and water, vertically; 
that is, there is no head. It is in the special relationship existent 
between the earth and mill-pond water that the energy lies, and 
not in either one, nor in both together. Literally, it lies in both 
apart. 

The same statements hold true of kinetic energy. There is no 
energy in a flying cannon-ball, for instance, unless there be 
something to stop it. It is only in its stoppage that the projectile 
can overcome resistance; and it is only a second mass-portion 
which can do any stopping. No mere geometric point, nor co- 
ordinate axis, can arrest a moving mass. To refer the motion 
of any mass to such a geometric base, as a measure of its energy, 
is an error so fundamental that it has adulterated our entire 
science of energetics. There is only one base of reference for 
the energy of any mass-pair, and that is its common center of 
mass. 

Kinetic energy, therefore, also consists of a relationship be- 
tween two masses. It belongs to neither mass alone, nor even to 
their aggregation alone, but to their aggregation when subdivided 
into separate portions by relative motion between them. 

In mechanical engineering these facts have long been lost 



MECHANICAL ENERGY 11 

sight of (although long known), for two reasons. In the first 
place, one of the two mass-portions, in engineering problems, is 
always the earth ; and the earth possesses so gigantic a mass, in 
proportion to that of any body of engineering magnitude, that it 
supplies an apparently fixed base of reference. Moreover, its 
great mass remains always constant, so far as engineering instru- 
ments can perceive, thus leaving the much smaller masses of our 
cannon-balls, railroad trains, etc., as the only variables. 

In molecular mechanics, however, it is not only quite possible, 
it is altogether probable, that the various mass-portions are of 
the same order of magnitude; or, at least, if there be all classes 
of magnitudes, that there are many of each class. Under such 
conditions it is of prime importance to remember that there are 
two mass-members, at least, in each unit of energy, and that one 
of them is just as likely to be a variable as the other. 

The second way in which the special conditions prevailing in 
engineering practice have warped its concepts of energy away 
from the truth is seen in the common idea of potential energy as 
synonymous with "up and down." To a student of purely mun- 
dane forces this is, of course, natural; but when one has studied 
true mechanics long enough to get away from the special condi- 
tions of the earth's surface, it is appreciated that potential 
energy refers only to "together and apart." There is no true 
"up" nor "down" in nature. It were well, for instance, as we 
stand by the brink of a Niagara, awed by the thunder of its 
action and marveling at even that slight modicum of its energy 
which the power-houses succeed in catching, to remember that 
almost beneath our feet the even more stupendous falls of the 
Zambesi, of more than twice the height and certainly equal 
power of Niagara, are thundering their watery masses in ex- 
actly the opposite direction from those before our eyes. 

Yet in either case the action is the same. The energy lies not 
in the water of either cataract ; it lies in the relationship existent 
between water and earth. And of this relationship no concept 
can be had by speaking of "up and down." It is the relative 
separation of earth and water, above the falls of either river, 
which constitutes the energy; and the only adverbs which will 
cover this idea are "together" and "apart." All portions of 
matter in the universe tend, by gravitation, to fall together, or to 
consolidate. All of them also tend, by centrifugal action, to fall 



12 ENERGY 

apart, or separate, or disintegrate. Sometimes, as on this earth's 
surface, the former tendency is overwhelmingly greater than the 
latter, and the latter is therefore lost sight of. But in many 
other places it is the centrifugal tendency which overwhelms 
and obscures the centripetal, so that the latter is forgotten. 
Such is usually the case in the so-called permanent gases, which 
tend always to expand indefinitely. 

But for our present purpose, viz. : A true statement of prin- 
ciples such as may fit all cases, all that is necessary is to keep 
both facts constantly in mind: that at all times and places in 
nature both tendencies, the congregative and the disgregative, 
are at work, in opposition to each other — sometimes one and 
sometimes the other prevailing. 

One of our national American humorists of a generation ago, 
in promulgating the manifold attractions of his world-famed 
"show," advertised the exhibition of one of the Siamese twins — 
"the only one which had ever been successfully separated." Not 
that the one had died, but that it had not yet been sepa- 
rated. The absurd humor of the remark never needed an ex- 
planation. And yet the two great twin forces of nature — the 
centripetal, or gravitational, and the centrifugal; the congre- 
gative and the disgregative — have ever been solemnly presented 
to our students of engineering, at different and disconnected times 
and places, as if each were the only one which had ever been 
separated from the other. They are taught as if each were an 
independent natural phenomenon. 

As a matter of fact, the action of gravitation is always and 
insistently to combine and unify mass; and if it had its way 
unhindered the universe would soon become a single solid of 
infinitesimal dimension and infinite density. Centrifugal force, 
on the other hand, always and insistently tends to separate and 
diffuse mass ; and if it had its way unhindered the universe would 
soon become a gas of infinite volume and infinitesimal density. 
But neither thing happens. So far as we can see, the mean 
volume and the mean density of the universe both remain con- 
stant. The simple explanation is that centripetal and centrifugal 
forces are always paired, in counterbalance. Sometimes one pre- 
vails, temporarily and locally, and sometimes the other. But 
neither is ever absent. Neither is ever completely free to act. 
The existence of either one alone is unknown in nature, is in- 



MECHANICAL ENERGY 13 

conceivable to the naturally taught mind, and should never be 
mentioned to the student as a natural possibility. 

Of these two fundamental mechanical tendencies let us con- 
sider first the centripetal one. 

It may be regarded as the prime fact of nature that, except 
when an excess of motion disguises the fact, all things are obvi- 
ously bonded together by a mutually attractive force. The "law" 
— or, better, the fact — of gravitation is simply the sublime condi- 
tion that each two bodies in the universe, of whatever sort, at 
whatever places and at all times, are drawn toward each other 
by a mutual tie of affection — an affection so true and unvarying 
that to liken it to mere human affection, which is always partial 
and fickle, were belittling it indeed. This tie can never be 
broken, although it can be stretched indefinitely. 

The old saw has it that it takes two to make a quarrel. Well, 
it takes two to make a bond of affection, just as well; and it 
takes two mass-portions to exert a gravitational attraction. As 
Newton defined the law, more than two centuries ago, the force 
between each pair of masses varies, first, directly as the product 
of their masses, and, secondly, inversely as the square of their 
distance. Stated mathematically this becomes 

Force = c MjMa-^, (1) 

wherein the M's represent the masses concerned, S their dis- 
tance of separation and c a constant. When the masses are 
measured in units, each of which weighs 32.16 pounds, on the 
earth's surface, and when S is stated in feet, the value of c, in 
order that the force shall also read in pounds, becomes about 
0.000,000,0343, or one divided by about twenty-nine or thirty 
millions.* 

If, in this formula, M^ should be stated as the mass of our 
earth and S as its radius, c M^-f-S^ would reduce to 32.16 
pounds ; or the gravitational force exerted by any unit mass at 
the earth's surface would be 32.16 pounds. If M^ and M2 were 
a pair of steel plates, one inch thick and about twenty-two feet 
square, hung up vertically in chains in contact, face to face, it 
would require a force of about one pound to overcome their 
mutual gravitational attraction and pull them apart. 

*These figures follow Professor Poynting, as quoted by Professor 
J. J. Thomson in Engineering (London), March 19, 1909, page 397. 



14 ENERGY 

This may seem like a very small force. But it is only because 
either one of the mated plates, pulling upon its brother, is a very 
small thing in comparison with the earth which is pulling upon 
them both. It is also to be remembered that the force increases, 
as the masses grow larger, by their product, and, as the distance 
grows smaller, by the reciprocal of its square. Thus it becomes 
plain how this force may become sufficient, upon occasion, to 
hold in position the gigantic heavenly bodies, on the one hand, or 
so to bind together the minute particles of steel, on the other, 
that a stress of 100,000 pounds per square inch cannot pull them 
apart. 

But force without space does not constitute energy. It is 
only as the separation alters that energy appears. Multiplying 
the force, therefore, by a small change in distance (dS, in the 
calculus) and integrating, there results 

Potential Energy = Ep = c M^M^ (-i - -^) = c M^M^ (-^^) (2) 

wherein So is any original distance of separation and S any 
other. As in Equation i, if S be measured in feet and M in 
units weighing g pounds the result will be a value for Ep in 
foot-pounds. 

If, further, the quantity S — Sc be given the symbol h, Equation 
constant, and if S — So be very small in proportion to S, then 
c Mj/SSo may be taken as a constant and given the symbol g. 
If further, the quantity S — So be given the symbol h, Equation 
2 becomes 

Ep = gM,(S-SJ = Wh (3) 

which is the formula for potential energy more familiar to engi- 
neers than the correct one. Equation 2. But Equation 3 now 
appears in its proper light, viz: as a mere approximation which 
suffices in accuracy under certain assumed special conditions only. 
The same general aspect of the situation applies also to the 
question of kinetic energy. If a moving body be accelerated, 
either positively or negatively, force is manifested and energy 
developed. But to this phenomenon the participation of at least 
two bodies is essential. No moving body can be accelerated 
except by reaction against a second body. We have Newton's 
own word for that. It is quite erroneous to say that any moving 
body possesses energy by reason of, and to the extent of, its 



MECHANICAL ENERGY 15 

visible motion. Relative motion between two bodies can be per- 
ceived visibly quite independently of their relative mass. We 
can measure the motion of distant suns from a base of observa- 
tion, to wit, the earth, so small that the sun in question would 
never perceive it if it hit us and wiped us out of existence. But 
the measure of the kinetic energy between two moving bodies 
depends altogether upon the mass of the point of reference. For 
a moving body possesses energy only to the extent that it can he 
stopped. And that extent is settled, not by the original body 
alone, but by the mass-relationship existing between motor and 
arrestor. 

If any mechanic doubts this statement, let him try to forge a 
bolt upon an anvil made of wet clay, which cannot bring its 
entire mass promptly to the job of arresting the hammer. Or 
let him try it even with a steel anvil, but one having a mass only 
equal to that of his hammer. He will find that his hammer, and 
the muscles which yesterday drove it with effective energy, to-day 
are powerless. The muscles are vigorous and the hammer-head 
lively; but action can be no greater than reaction. As the anvil 
reacts, so only may the hammer act. Motion may be there, but 
not necessarily kinetic energy. 

These ideas may be reduced to mathematical expression by 
starting with the empirical equation of motion. 

Force = Mass x Acceleration. (4) 

From this there results, by algebra which need not be reproduced 
here, the following fundamental expression for the kinetic 
energy of any mass-pair : 

1 M M 

wherein the M's are the two mass-portions (measured in units 
weighing 32.16 pounds at the earth's surface), the V's are the 
original and final velocities, in feet per second, respectively, and 
Ek is the energy in foot-pounds.* 



* The reader may be assisted to connect these fundamental formulae 
for energy with his more familiar engineering concepts by the following: 

Let Fig. A represent two bodies, Mi and Ma, having a common center 
of mass at C. Let the bodies possess a motion toward or away from C 



16 ENERGY 

If Ml be very large in comparison with M^ the fraction re- 
ducts to approximately the value Mg. If, further, Vo = o, the 
expression becomes 

Ek = ^MV^ (6) 

which is the special approximation which the engineering pro- 
fession generally regards as the true fundamental equation for 
kinetic energy. 

So long as the mind confines itself to engineering problems 
this special approximation serves as well as the exact expression. 
But the text-books certainly ought never to leave it to the maga- 
zine press to inform the student that it is a special approximation, 
instead of the true article. And when this special approximation 
is carried into the problems of molecular energy, where it is 
applied as the sole available concept of kinetic energy, this bit 

of V, and Va respectively. Then, from the law of conservation of mo- 
mentums, 

MiVi = M2V2 (a) 

and Vi + V2 = V (b) 

wherein V is the relative velocity between the two bodies. 

If this relative motion be opposed by a force of magnitude 
Miai (= Msaa), acting upon each mass and reacting upon the other, it 
will be destroyed at a rate of negative "acceleration" equal to 

A = ai-l-aa (c) 

wherein at is the change in velocity per second measured between Mi and 
C, as is that measured between Ma and C, and A is that measured between 
Ml and Ma. 

When the motion has been entirely overcome the opposing force will 
have been overcome by the body Mi through the distance § Vi, and by 



... 1 

a 




c 

M 

FIG. A. 

the body M3 through the distance | Va, or over a total distance of ^ V. 
The work performed will have been equal to the force times the distance 
in the case of each body, or, together, 

Work = MiVi (I Vi) +M2V2 (i va) (d) 

From Equations a and b, VMa . , 

^' = M, + M, ^'^ 

, VM. „^ 

The sabstitution of Equations e and f in Equation d gives 

Work= i ^f'^[\ V^ (g) 

^ Mi + M2 ^^^ 

for the special case where the relative motion is completely destroyed. 



MECHANICAL ENERGY 17 

of careless neglect becomes an egregious, fundamental error, 
which has misled many an able mind which lacked the time 
needed to dig out the straight of the matter. 



The significance of the above, aside from some most im- 
portant conclusions which will be drawn from it in later pages, 
amounts to this : 

1. Energy exists only between bodies, and never in them. 
No single body, by any quality assignable to it as a unit, can 
ever possess energy. 

2. The energy frequently spoken of as internal to a "single'' 
body implies that the body in question is not a single unit, but a 
swarm of tiny particles, each separate from its neighbors, yet 
too small to be seen separately, between which exists the energy 
said to be ''internal" to the body. 

3. The relationships between mass-portions which constitute 
energy may be either of two sorts, viz : j/jac^-relationships and 
mo/?on-relationships, called potential and kinetic, respectively. 

4. Neither relative space nor relative motion themselves con- 
stitute energy, but only change in either space or motion. There- 
fore, every true expression for energy must contain the differ- 
ence between a greater and a lesser value for space or motion, as 
the case may be. And it is a general fact of nature that nowhere 
is the smaller measure of either space or motion ever known to 
become zero. 

5. Every true expression for energy must contain, for its 
mass-factor, the arithmetical product of the two quantities of 
mass concerned. 

6. The elementary or unit mass factor in energy is not a 
unit of mass, as is now universally taught, but the unit mass-F air; 
that is to say, a pair of mass-portions, each member of which is 
one unit of mass. 

7. The ability of different mass systems to embody energy 
is proportional, not to the total mass of each, but to the square 
of that aggregate mass. This is necessarily true of space-energy, 
but may or may not be true of kinetic energy, according to the 
nature of the motions contemplated. 



CHAPTER 11. 

Free and Vibratory Energies. 

In the preceding article energy was defined as a change in 
either the space-relationship br the motion-relationship between 
two or more bodies. It is necessary now to see what sorts of 
space and motion relationships are possible in nature. 

Let Fig. I represent an ordinary pendulum attached to two 
A-frame supports on the earth's surface. Here exists an energy- 
system in which both space and motion relationships occur. 
Moreover, assuming the pendulum to be in motion, there is a 
constant change of both. Finally, there are present the two 
separate mass-portions which were stated in the preceding paper 
to be essential, between which the energy is embodied. One 
mass-portion is the pendulum-bob M^ and the other is the 
earth M2. 

If M^ be held stationary at A the system exhibits space- 
energy only. If it be released at A, however, the pendulum 
swings toward C, speeding up as it goes. But at C it has reached 
its maximum velocity, and beyond that point it slows down until 
B is reached, where it stops, reverses and repeats the process in 
reversed order. In this simple and familiar phenomenon is ex- 
hibited one of the broadest and most impressive principles of 
natural action, viz : 

THE CONSERVATION OF ENERGY. 

For if the space-energy lost between A and C be carefully meas- 
ured, and also the motion-energy gained, the two will be found 
to be equal. Expressed mathematically, 

wherein the a-subscripts refer to conditions at either A or B, and 
the c-subscripts to those at C. This is the fundamental equation 
for energy-transformation. 

Fig. I, however, does not represent an ideal or perfect 
system, because the connection between the two masses is estab- 
lished by means of a nominally flexible cord or hinge, which in 

18 



FREE AND VIBRATORY ENERGIES 19 

actuality always involves friction. Nor may it be inferred from 
this that the words ideal and perfect refer, as they usually do 
when used in connection with heat-engines, or gases, to some 
imaginary and impossible conditions, never found in nature. 
For in mechanics, while there is no known instance of force 




FIG. I. 

being transmitted from body to body hy contact without some 
loss of energy in friction and impact, yet it is common for such 
to occur by action at a distance. Indeed^ there exist no two 
bodies in the universe, according to Newton, between which force 
is not thus perfectly transmitted, by gravitational attraction, and 
between which motion cannot occur under forceful control, yet 
without friction. This fact is the very foundation of our entire 
science of mechanics.* 



*This statement takes entire cognisance of the resistance to the motion 
of celestial bodies through interstellar space which has been revealed by- 
modern astronomy. The fact of this resistance upholds all' the positions 
taken here and in later pages, as to our fundamental concepts of true 
mechanics, celestial or molecular. The laws of Newton and Kepler, and 
the fundamental mechanical principles which have been deduced therefrom, 
are all based upon the assumption that the celestial bodies move through 
matterless space. They hold true only in that case. Indeed, our only 
idea of a "body" is a portion of matter separated from other portions of 
matter by true space, in which exists no matter. Later in these pages it is 



20 ENERGY 

The use of this gravitational action at a distance to link two 
bodies together into an energetic system similar to Fig. i is shown 
in Fig. 2, as a diagram of an energy-transforming system which 
would remain in operation indefinitely, with its energy perfectly 
conserved ; because it relies solely upon "action at a distance," or 
force transmitted without friction. In Fig. 2 one of the masses, 
Mg (the earth, if you please), is shown pierced centrally with a 
great well-hole — much as Captain Symmes imagined the earth 




FIG. 2. 

to be from pole to pole, in his long-ago famous "Symmes Hole" 
theory. Above the earth's surface and in line with this hole, as 
at A, is suspended the body M^. Upon its release it will vibrate, 
quite as did the M^ of Fig. i, on either side of the point of 
closest approach to M2 at C. 

To this system Equation 7 applies perfectly. In this case, too, 
occurs good illustration of how the lesser distance of separation 
Sc never becomes zero, although at the point C the geometric 
centers of the two spheres happen to coincide. But in energetics 



stated, as a broad natural empiricism, that there is no such a thing as true 
"space" devoid of matter — that some degree of mass, pressure, temperature, 
etc., exists everywhere. This is upheld by the fact of resistance to celestial 
motion showing that the celestial bodies move through, not space, but 
attenuated matter, probably interspersed with small solid bodies. It will 
also be pointed out that there exists in nature no such a thing as true 
matter; that is, matter devoid of space. 

All this does not affect the fact that we are now two centuries gone on 
a course starting from the concept of true matter and true space, as con- 
trasted absolutes. Every boy's experience, every student's training in 
elementary mechanics and every engineer's instinctive judgment are based 
upon this same idea, which is also the foundation of the Newtonian 
mechanics. The writer is merelv insisting that, since the doctrine of all 
mechanics as founded on the Newtonian concepts, and of heat as being 
"a mode of motion," is promulgated universally in all engineering schools, 
its consequences must be faced with consistency. 



FREE AND VIBRATORY ENERGIES 21 

it is only mass which counts, not geometry. Although the geo- 
metric centers may coincide at C, the two masses do not. After 
Ml enters the geometric boundaries of Mg their real separation 
decreases only gradually, until at C it is a minimum, but not zero. 

It happens, however, that the case illustrated in Fig. 2 never 
occurs in nature. It is not that the major mass-portions of the 
universe neglect to possess convenient openings for the passage 
of the minor mass-portions through them. The trouble is that in 
nature the chances are infinitely against any pair's ever possessing 
a motion, when in the condition A, which is directly alined 
with their mutual centers. The thing is conceivable geo- 
metrically, but it is even less than probable. When the natural 
causes of different forms of motion are investigated, it will 
appear that such motion is impossible. Motion developed nat- 
urally, rather than in the imagination, will be directed at some 
appreciable angle with the mutual axis, as at A in Fig. 4. 

In any natural case the mutual motion will be likely to assume 
the form portrayed in Fig. 3 — to consider the simplest case first. 
Here the bodies are shown as revolving about each other, and 
also about their common center of mass at C. Each body de- 
scribes an elliptical orbit about its mate, and also about the 
point C. For a moment's consideration will show that, whatever 
form of orbit one body follows around the other, the other must 
likewise describe about the one ; and if the mass-center C be 
regarded as the fixed center, each body will describe about it 
similar orbits, having radii inversely proportional to the mass in 
question. 

The situation is much as if a boy fastened a large cannon- 
ball on one end of a stick and a smaller one on the other. He 
can then twirl the stick by holding either end in his hand, re- 
garding that as fixed while the other end moves, or he can hold 
the middle portion of the stick in his hand, at the center of 
gravity, and twirl both ends at once about that. But in the case 
of the boy and the stick, his hand is attached to the earth, and 
m.ay be regarded as fixed; whereas in the case of the two free 
mass-portions, if we are to study their own interaction, inde- 
pendently of all other masses, there is no fixed base to refer 
anything to. Should the boy throw the stick into the air, how- 
ever, for a brief period it would act freely as an independent, 
free mass-system, and during that time the two cannon-balls 



22 



ENERGY 



would revolve, each about their common center of mass. For 
that reason it is proper to consider the energetic action of the 
members of a pair only in reference to their common center of 
mass. 

In any such an elliptic orbit the condition of greatest separa- 
tion, as at ab, is called the apastron of the orbit, while that of 
least separation, as at AB, is called the periaston. At apastron 
the energy existent between the two is chiefly space-energy, but 
there is also a little motion. At periastron the energy is chiefly 




FIG. 3. 

motion-energy, but there is also a little space present. The 
equation which connects mathematically these maxima and 
minima of space and motion is Equation 7 ; and in it, in all truly 
natural cases, neither Vo nor So may ever be regarded as be- 
coming equal to zero — as will be made plain as the argument 
proceeds. 

For Fig. 3 — or, rather, its more general form, Fig. 4 — repre- 
sents the only true element of mechanical action. For any such 
an element, in order to be an element, must be absolutely "free." 
That is to say, it must be considered independently of all other 



FREE AND VIBRATORY ENERGIES 23 

masses. Yet it must be capable of containing energy. There- 
fore, it must be, not an ultimate or indivisible unit of mass, but a 
mass-pair, an elementary mass-pair. 

All action between solid bodies in contact, on the other hand, 
such as is familiar to all engineers in their machines, is not 
purely mechanical. It is always partly thermodynamic, in so far 
as heat is being constantly developed by friction, and partly a 
special case of pure mechanics, in that the body is "constrained" 
rather than free; that is, it is handling energies which are 
transient through it from without, which are independent of its 
own mass, and which are ultra-complex in their nature. 

Fig. 4, on the other hand, is entirely general. It displays 
every possible form of pure and elementary mechanical action 
between two bodies, supplied with any original store of relative 
space and relative motion whatever, as at A; and it introduces 
no foreign element of dependence upon any other mass-system 
or form of energy whatever. Nor does it introduce any un- 
natural assumptions. Without stopping now for the proofs, it 
may be said that any such a case must resolve itself into the 
mutual revolution of the bodies about each other in an orbit 
which follows some of the plane conic sections — either the 
hyperbola, the parabola, the ellipse, the circle or the straight line. 

Further, it can be shown that if the original energetic condi- 
tion of the pair at A M^ be known, by knowledge of the dis- 
tance d between the two, the velocity v of their relative motion, 
the angle <}> existent between d and v, and the two masses M^ and 
M2, then the nature of the orbit is known, and also its dimen- 
sions. Both are best expressed in terms of the distance D be- 
tween the two when separated by a radius normal to the major 
axis XX' of the orbit, the velocity U at that point, and the angle 
a between D and U. The mathematical relationships between 
all these quantities will be discussed later. 

Of all these apparently varied forms of motion only two, the 
hyperbola and ellipse, are probable forms. For between any two 
masses, at any given initial distance, there may be an infinite 
number of directions and magnitudes of velocity which would 
result in hyperbolic motion, and another infinite number which 
would result in elliptic motion. But there is only a single direc- 
tion of motion which would result in a straisfht-line orbit, such 
as that of Fig. 2 ; and there is only one other direction of motion, 



24 



ENERGY 



with a particular velocity into the bargain, which would result 
in circular motion. A parabolic orbit could likewise result from 
only a single direction of motion, coupled with its proper mag- 
nitude. Therefore the mathematical chances are infinitely in 
favor of all motion in the universe being either elliptic or hyper- 
bolic in form. 

The possibilities, however, are not even so complex as even 
this statement might indicate. Speaking mathematically, there is 
for all of these forms of orbit but a single, basic equation. This 
equation must involve some factor expressive of radial distance 
between the two, and also some factor expressive of either the 




FIG. 4. 

two quantities of mass or their periodic velocity of revolution. 
These will define the general magnitude of the system. But 
besides these factors there must also be present, as an indicator 
of the form of orbit, another which is called the eccentricity. 
And the eccentricity of orbit, it will be found, is a factor of 



FREE AND VIBRATORY ENERGIES 25 

fundamental importance in the determination of whether energy- 
transformation is to occur or not. 

Without attempting to enter into any discussion of the mathe- 
matical definition of eccentricity, which is usually given the 
symbol e, the following characteristic facts should be noted care- 
fully, as indicating sufficiently for our purposes its general 
nature : 

1. If the eccentricity be zero the orbit is a circle; 

2. If the eccentricity be greater than zero, but less than 
unity, the orbit is an ellipse; 

3. If the eccentricity be unity the orbit is a parabola ; 

4. If the eccentricity be greater than unity, but finite, the 
orbit is an hyperbola ; 

5. If the eccentricity be infinite the orbit is a straight line. 

Of all possible cases, therefore, the circle constitutes one 
extreme and the straight line the other. Mathematically speak- 
ing, they constitute mathematical limits, with the chances infi- 
nitely against their ever being attained in nature. But, speaking 
naturally, this means that they never occur. Both zeros and in- 
finities are unimaginable, as natural phenomena. They have 
never been observed, and, so far as the human mind may 
predict, they never will be observed. It is one of the heaviest 
indictments to be brought against our present methods of teach- 
ing natural science that its text-books are so filled with reckless 
use of zeros and infinities. It is not that zeros and infinities 
should never be employed, but that they should always be speci- 
fied, before using, as natural impossibilities — as is always done, 
for instance, in specifying the exclusion of friction, thermal con- 
duction, etc. 

As for the parabolic orbit, that plainly stands as the dividing 
case between the ellipses below and the hyperbolas above. It 
itself, like the circle and the straight line, is infinitely improbable 
of occurrence, in any permanence of form. But whereas the 
chances are infinite against truly circular or straight-line motion 
ever being attained, even instantaneously, the condition of para- 
bolic motion must be at least crossed, and therefore existent in- 
stantaneously, in transition from elliptic to hyperbolic motion. 
For the parabolic condition Is like a dividing line between two 
areas: having no dimension, substance or reality itself, it must 



26 ENERGY 

nevertheless be encountered and crossed by substantial realities, 
in their passage from one territory to the other. 

But, it will naturally be asked, what have all these astronom- 
ical illustrations of *'free" bodies to do with engineering me- 
chanics, when no engineer or mechanic ever saw anything in his 
work which acts in that way ? The answer is that, in spite of the 
strangeness of the idea, this is the only way in which any portion 
of matter ever really does act, when it acts purely mechanically. 
In some departments of mechanics, chiefly in gunnery, the pro- 
jectiles which form the chief subject of study do come very 
nearly to acting in this way. The only assumption which has to 
be made, in reducing their ordinary action to a true, or *'free," or 
natural basis, is that the friction of the atmosphere be absent. 
Then their paths are commonly said to be parabolic. But this is 
a statement which is only approximately true. It is made be- 
cause it is simple and easy to assume that the lines of gravita- 
tional force which radiate from the earth's center are, for small 
portions of its surface, virtually parallel ; and also because the 
result is not far enough wrong to upset calculations. 

But for the special purposes of our argument it is far better 
to keep in mind the exact truth — which is simple enough so long 
as we do not try to make any detailed calculations — namely, that 
all cannon-balls, base-balls, foundry-drops, etc., are in reality 
following elongated elliptical orbits passing closely around the 
center of the earth. But of this orbit the only portion which 
we can see is the apastron tip. The remainder of the orbit 
never is traversed, because collision with the solid surface of 
the earth breaks up the phenomenon, in a dissipation of energy 
in thermodynamic, rather than mechanical, action.* 

But even to get such a slight peep as the above at pure me- 
chanical action on the surface of the earth, we have had to 



*The foundry-drop is ordinarily regarded as falling toward the earth in 
a straight ver'^ical line. Yet, in reality, as it leaves the latch it possesses 
a tangential velocity, parallel with the earth's surface, of over fifteen hun- 
dred feet per second, due to the earth's rotation. If we could imagine the 
mass of the earth as suddenly concentrated at its center, or within a sphere 
about ten miles in diameter, the foundry-drop would be free to describe an 
elliptic orbit passing around the center of the earth at a distance of about 
seven miles, and exhibiting at periastron a tangential velocitv of 170 miles 
per second. Midway toward the center of the earth this ellipse would 
spread out to a conjugate diameter of over one hundred mii'^<?. Yet this 
illustration is the nearest approach to rectilinear motion which we can 
produce in the engineering arts ! 



FREE AND VIBRATORY ENERGIES 27 

make the never true assumption that the friction of the atmos- 
phere were absent. Yet in all the mechanical actions which are 
even more familiar to the engineer, such as the interaction of 
solid machine-parts, see what even more wholesale assumptions 
have to be made, in order to weed out the non-mechanical 
phenomena of friction and impact and get a glimpse of the pure 
mechanics behind them ! It is scarcely worth while to try. And 
yet it is none the less true that there is never a hammer-head, 
piston or shuttle started into motion that it does not try to 
follow an elliptic orbit about the earth's center; from which it 
is constrained only by constant supplies of energy from without, 
in the form of solid forces not dependent upon the masses trans- 
mitting them, which forces and energies are called "transient." 

For taking the time to study and understand these free or 
natural tendencies of mass, which get so little chance to display 
themselves here upon the earth's surface, there are two reasons. 
One of these reasons is the fact that they constitute the only 
true mechanics, from which all engineering happenings are but 
special departures and for which only approximate statements 
can be made. It is therefore the soul of true education to teach 
these exact truths first, displaying their useful applications after- 
ward in their proper aspect. 

The other reason is that no concept of heat as a mechanical 
phenomenon can possibly be attained without them; for in the 
mechanical interaction between the particles of a body, by which 
we must now attempt to explain not only heat, but surely also 
chemical action and probably also electrical and magnetic ener- 
gies, there can enter no friction nor impact. The action must 
be purely mechanical. There can be no dissipation or degrada- 
tion of energy into heat, because it is heat itself which is being 
considered. Nor, as stated in the last paper, is it any explanation 
of the puzzle to specify that the particles shall be perfectly 
elastic; because perfectly elastic matter is unknown in nature. 
It is only space, devoid of solid contact, which is perfectly 
elastic in nature ; and the first task in the comprehension of 
heat, therefore, is to comprehend thoroughly this "free" depart- 
ment of natural action, in which friction and impact are unknown 
and unimaginable. 

The task of the theorist, in fact, is not so much to explain 
action at a distance in terms of action by contact, as is so often 



28 ENERGY 

assumed, but fairly the reverse. The obscure and intricate hap- 
penings, which, in our ignorance we lump together under the 
convenient blanket- term "contact," involving always interchanges 
of pressure and heat, and often mechanical, chemical and elec- 
trical energy, none of which we understand, can be explained 
simply and clearly only when they are reduced to terms of 
action at a distance. For the latter demands no "explanation," or 
reduction to terms of something else. It is beautifully simple, 
having been defined in an elementary algebraic formula by 
Newton two centuries ago. It is complicated by no questions of 
elastic pressure, thermal or electrical conduction, or chemical 
interaction, varying interminably with each new case of "con- 
tact." Centripetal gravitation, like its mate, centrifugal force, is 
one of the basic facts of the universe — more basic even than 
matter, the existence of which we infer from its gravitational and 
centrifugal action — and forms no proper food for further analysis 
until it shall appear to us in much more intricate guise than that 
which we inherit from Newton. 

Our sole duty here is to recognize that "contact" is merely a 
convenient name for impact and friction, when we are discussing 
mechanics, for thermal conduction when we are discussing heat, 
for electrical conduction when we are discussing electricity, etc. 
''Action at a distance" implies merely the absence of any of these 
energetic transformations. 

These, then, are the elementary laws of motion, when viewed 
from the standpoint of energetics rather than kinematics. Since 
they appear to be quite different from the laws of motion of 
Newton, with which every student is familiar, while apparently 
of an elementary importance equal to those of Newton, it is 
important to note that they are in reality the laws of Newton, 
but stated in combination with each other and with the laws 
discovered by Kepler some sixty years before Newton enunciated 
his law of gravitation in 1680. Newton's laws of motion, in 
their familiar form, constitute one exceedingly simple form in 
which the elements of mechanics may be expressed. The trouble 
with their form is that, in order to get each statement into its 
simplest possible form, a set of premises has been assumed 
which is peculiar to that particular statement only, and which 



FREE AND VIBRATORY ENERGIES 29 

not only never occurs in nature, but which is directly incon- 
sistent with some other of this same set of laws. 

For instance, one of these laws of Newton asserts that "a 
body once in motion will continue in unchanging straight-line 
motion until interfered with by a force." But Newton's own 
law of gravitation declares that no body in the universe may 
ever dissociate itself so remotely from all others as to be free 
from interference by forces, and by an infinite number of forces 
at once. Therefore, unchanging or straight-line motion can 
never occur in nature. For some special problems of mechanics 
it has been useful to assume that it could. But the wholesale 
manner in which this special and temporary assumption has been 
permitted to usurp the 'place of a fundamentally correct and 
permanent principle of nature has undermined our entire under- 
standing of the problem of energetics. 

Again, for instance, "a body subject to a constant force will 
experience constant acceleration." But again, Newton's own 
law of gravitation declares that force can remain constant only 
so long as the separation between the bodies remains constant. 
But this is impossible, motion being assumed at all, except when 
the motion has the form of a circular orbit, at constant distance ; 
in which case the motion and force would be at right-angles with 
each other, and mutually independent. In all other forms of 
motion the force must always be varied by the motion which it 
itself produces. Therefore, constancy of acceleration, whether 
positive or negative, is unknown in nature. 

To understand properly Newton's laws of motion, therefore, 
they must be coupled with each other and with the laws of 
Kepler — upon which latter, indeed, they were founded. Kepler's 
laws are three in number, viz : 

1. The natural path of free motion between two masses is 
one of the plane conic sections.* 

2. The area swept over by the radius-vector connecting the 



*Kepler, working with the planetary orbits alone as his material, and 
quite as a pioneer, stated this law originally as including only the ellipse. 
It is later learning which has broadened the statement to include all of 
the conic sections. A very simple and elegant proof of this law — depend- 
ing, however, upon Kepler's Second Law — was published by Mr. Immo S. 
Allen, of the London Institution (Finsbury Circus, London, E. C), in the 
Scientiiic American of July loth, 1909. 



30 ENERGY 

two is everywhere equal for equal periods of time; or the "area! 
velocity" is constant. 

3. The square of the orbital period (or time) of revolution 
is proportional to the cube of the major axis of the ellipse, 
divided by the sum of the two masses. 

The third of these laws again illustrates the way in which 
fundamental error has been permitted to creep into the standard 
college text-book. Of all of the ordinary text-books in astronomy 
which the writer has happened to enter — not to find fault, but 
in a sincere effort to straighten out this tangled question of 
"what is energy" — only one mentioned the sum of the masses at 
all, as a factor in Kepler's third law. All others omitted it, 
making of the expression for the third law a special approx- 
imation as erroneous as is the formula ^ M V^ for kinetic 
energy. 

Indeed, until this single case was discovered, the writer was 
quite puzzled by the situation; for, according to all he knew of 
the elements of mechanics, the sum of the masses ought to appear 
in the statement of Kepler's third law. And yet here was 
standard text-book after text-book which made no mention of 
them ! Was all that he thought he knew nonsense, or where 
else was the trouble? In his quandary his heart went out to all 
those unfortunates who may have made serious attempt to under- 
stand the general principles of energetic action from the me- 
chanics taught in the colleges as a foundation. 

If, now, all the laws of Newton be kept in sight at once, and 
those of Kepler with them^, the elementary principles of all 
mechanical action may be restated as follows. In their literal 
form they apply only to an energetic system consisting of a 
single pair of bodies. In the sense that every portion of the 
natural universe may be — and, according to any philosophy 
founded on the Newtonian mechanics, must be — considered as 
made up of a large number of such pairs, with its distances, 
forces and motions all reducible to an equal number of com- 
ponents, one for each pair, they are universal in their appli- 
cation. 

I. Everywhere is space. No two bodies may be conceived 
as coincident, nor any one body as occupying zero space. The 
"occupation of space" has always constituted the fundamental 
definition of "matter." 



FREE AND VIBRATORY ENERGIES 31 

2. All space is relative, measurable only between bodies. 
Absolute space is as inconceivable as is absolute lack of space. 

3. Everywhere is force. Freedom from finite force is un- 
known. No two bodies may ever become so widely dissociated 
as to reduce their mutual attraction to zero, nor so closely coin- 
cident as to raise it to infinity. 

4. All force is relative. It exists only between bodies, and 
may never be imagined as exerted absolutely, independently of 
mass. 

5. Everywhere is motion. Absolute rest, or fixity, is un- 
known and inconceivable. 

6. All motion is relative. Motion is measurable and con- 
ceivable only between the members of a related pair. No single 
body, independently of all others, may possess motion. Absolute 
motion is as inconceivable as is absolute rest. 

7. Constancy of either space, force or motion is unknown in 
nature. Space varies force, force varies motion, and motion 
varies space, all the time, between any and eveiy two free bodies. 

8. Neither straight-line nor circular motion is known in 
nature. The only path of motion which is natural, rather than 
imaginary, hypothetical and superstitious, is either the ellipse or 
the hyperbola^ with velocities varying as stated by Kepler and 
forces varying as stated by Newton. 

9. In any energetic system the primary fact is the con- 
stancy, or indestructibility and non-creatability, of its mass. The 
Principle of the Conservation of Mass, discovered first and 
needed first, should certainly receive the title of First Law of 
Energetics. 

10. In any such a natural, free system there occurs period- 
ically, at each revolution, a reversed transformation of energy, 
from space to motion form and back, under the Principle of the 
Conservation of Energy. Discovered only in 1837 — after much 
preliminary investigation and partial knowledge — and not yet 
fully understood, this great natural principle is properly to be 
entitled the Second Law of Energetics, not the "First," as it 
is now called. 



CHAPTER III. 

The Mean Energetic Condition and the Energy-Fund. 

In the last paper attention was called to the fact that all 
"free," or natural systems of mechanical action might be repre- 
sented by some one of the conic sections, such as were exhibited 
in Figs. 3 and 4. Of these, for the present purposes of dis- 
cussion, the elliptical orbit, as shown in Fig. 3, will suffice as an 
illustration. 

In elliptical motion, as w^as pointed out in that paper, there 
occurs a periodic energy-transformation at each revolution of the 
bodies about each other. At apastron space is a maximum and 
motion a minimum; at periastron the re.verse is true. And at 
every point between these extremes the conservation of energy 
is maintained ; the amount of either form of energy lost, below 
the maximum, is made good in the other form. 

From these facts it is readily inferred that there must exist 
between the extremes a and A some intermediate point where 
this transformation of energy from space-form to motion-form, 
or the reverse, is just half accomplished. Such a point would 
constitute a true energetic mean between the two extremes. At 
that point half of the total range of potential energy will have 
been converted into or from the kinetic form, and half will yet 
remain to be converted. 

Let us indicate the distance of separation between the two 
bodies when in this mean energetic condition by D, and their 
relative velocity by U. Then there must result, from Equation 7 
for the conservation of energy, 

1 1 M M 1 M M 

-Ic M,M, (-^ ^) = c M,M, (-| ^) (9) 



and 



These equations readily reduce to 

J. 
2 



U^ = 4- (V^+Vo^) (10) 



32 



s= 


= a(H-e) 




So = 


= a (1-e) 




2 a 


Ml-e^) 


b2 




2a 


a 



THE MEAN ENERGETIC CONDITION 33 

CO 

and D = 2-^^ (11) 

In the case of elliptical orbits it is usual to represent half the 
major axis by a, half the minor axis by h and the eccentricity 
by e. In that case 

(12) 
(13) 

and D- ^^o" ^' =^ (14) 

b^ 
But — is one-half of the latus rectum of the ellipse, or the 
a 

diameter through either focus at right angles to the major axis. 
That is to say, the mean energetic distance D is the vector or 
radius joining the two bodies when they are situated directly 
at right-angles to the axis joining apastron and periastron. 

In short — and this seems to be most important — the energy- 
transformation which is always a part of the revolution of two 
bodies about their common center of mass, is a function of 
angular, not linear, motion. No matter how eccentric the ellipse 
may be, it is always true that in each quadrant of its motion the 
energy is just one-half transformed — from extreme space or 
extreme motion to the mean energetic condition, or back. 

There is ^ne case in our own solar system, for instance, 
where one quadrant of the elliptic orbit, that from apastron to 
the mean energetic condition, occupies about four hundred years 
and covers a distance measurable in hundreds of millions of 
miles. Yet the next quadrant, from the mean energetic condi- 
tion to the extreme energetic condition nearest the sun, trans- 
forming an equal quantity of energy, occupies only a little over 
an hour and covers a distance measurable in thousandths of the 
other. And there may be, of course, even more extreme illus- 
trations of eccentricity of orbit than this. 

In Fig. 3 this mean energetic position is shown at DD', and 
in Fig. 4 at BMg. In Fig. 4 the mean energetic velocity U is 
shown as maintaining the angle a with the vector, or latus 
rectum, D. It will be convenient to note, concerning this angle a 
for future purposes, that 



34 ENERGY 



e= — Va^-b^ =cotan a (15) 

a 

These statements and arguments, although they are expressed 
most simply in terms of the ellipse, could be established as apply- 
ing equally to any of the conic sections, such as are shown in 
Fig. 4. In the case of the parabola and hyperbola, which have 
no apastron ends and no definite major axes, the case is compli- 
cated (as will appear later) by the presence, within the pair of 
mass-portions, of two funds of energy — the one which has just 
been discussed and another. For the present it will be quite 
sufficient to the student to accept on faith the statements that 
the discussion is founded upon principles which are general in 
their character, applying to all possible cases. 

In general, then, if the original energetic relationship of any 
pair of masses whatever, such as M^ and Mg of Fig. 4, be 
known — by data as to their masses, their distance d, their ve- 
locity v, and the angle cf> which the latter makes with the vector d 
— all the conditions of the orbit and the energy-fund embodied 
by the pair are known. The all-important question as to the 
amount of energy thus embodied, which is a very difficult one, 
will be resumed at a later point. What is of more elementary 
concern at present is the form of the energetic action; and this 
depends primarily, it is obvious, upon the eccentricity of orbit. 

It is worth while to repeat, in a briefer form and with in- 
clusion of the angle a, the list of the different (mathematically) 
possible types of orbit which was given in the preceding paper. 

1. If 6 = 0, a = 90° and the orbit is a circle. 

< 90° 

2. If e< 1, a .ro and the orbit is an ellipse. 

3. If e = 1, a = 45° and the orbit is a parabola. 

4. If e > 1, a< 45° and the orbit is an hyperbola, 

5. If e = a, a = 0° and the orbit is a straight line. 

It is therefore most significant that, throughout all this wide 
variation in values of e and x, and in diversity of form of 
orbit, the mean energetic condition should remain consistently 
at the position normal to the orbital axis. Whatever may have 
been the original distance of separation from which the two 
bodies fell together, or whatever may have been the angle swept 
over by the vector between the positions A and B, or whatever 
may be the speed and propinquity at which the bodies pass each 
other at periastron, it always holds true that, once the mean 



THE MEAN ENERGETIC CONDITION 35 

energetic condition is reached and one-half the energy-trans- 
formation accompHshed, at B, a further swing of just one quad- 
rant is consumed in completing the other half of the energy- 
transformation, to periastron at P. After that a second quadrant 
is consumed in reversing this energy-transformation, to the mean 
energetic condition again, at B' ; after which the cycle ends with 
a reversal of the angle AMgB, whatever it may have been — a 
quadrant in circle or ellipse, or less than a quadrant in the 
parabola or hyperbola. 

Should the line BE' of Fig. 4 be considered as representing a 
plane normal to XX', and 90° — a as the angle of incidence 
thereto, the angle of reflection therefrom, at B', will always be 
its equal. 

The Energy-fund: Radial and Tangential Energies. If 
it be true that the original position and motion of the pair 
at any original condition, such as A, Fig. 4, defines all the 
conditions of the orbit, it should be possible to define in terms 
of them the total fund of energy existent in the pair. Mathe- 
matically speaking, this is possible — so far as it is possible to 
define an energy-fund in terms of any premises at all. But the 
equations which connect any original condition, such as A, with 
the conditions B, P, etc., are so cumbrous in form, when com- 
bined, that it is not practicable thus to define the energy-fund in 
terms of any original point. It must suffice, instead, to know 
that the connection between A and every other point in the 
orbit is rigid and exact. It is quite sufficient for present pur- 
poses to investigate the energy-fund in terms of the conditions 
B and P only, as bases. 

The kinetic energy visible at B In the velocity U is divisible 
into two components, the radial and the tangential, respectively. 
To each of these must correspond a respective fund of energy, 
radial anl tangential in its nature. Now, these two funds of 
energy bear a marked contrast with each other, in many re- 
spects ; and as this contrast runs through the entire question of 
energetics, It Is worth while to discuss it somewhat carefully. 

In the first place, space-energy can, of its very nature, exist 
only radially. It is impossible to think of pure separation between 
two bodies — which, if they be truly single bodies, must be 
regarded as geometric points at their centers — ^in connection with 
any idea of direction of separation. Moreover, it is not only 



36 ENERGY 

that direction of separation is unthinkable in this connection ; it 
would have no effect if it were conceivable. For the force of 
gravitation operates equally in all directions, and radially only; 
and as space-energy is based directly upon this force, it too must 
be regardless of angular direction. 

Now this energetic force of gravitation works always in one 
direction, of the two directions possible within each radius. It 
always urges the two bodies together. Urging the two bodies 
apart, at all times, is the centrifugal force; and this force is a 
function of the tangential motion-energy. The two forms of 
energy, spacial and kinetic, are therefore in energetic counter- 
balance, just as much as their forces are in dynamic counter- 
balance. When the space of separation becomes greater than 
normal, it prevails over the motion-energy and forces the two 
bodies into greater propinquity. When the motion becomes 
greater than normal — and this is always the result of the action 
just noted — it prevails over the gravitational attraction and 
forces the bodies apart. Thus these two forms of energy vibrate 
against each other, in stable equilibrium, about the mean ener- 
getic condition as a center. 

Now this centrifugal force, although itself radial in direction, 
is developed only by tangential motion; and it is in permanent, 
not vibratory, equilibrium that the tangential component of the 
velocity U finds itself in counterbalance with the gravitational 
force. To explain, let us consider first the case of purely cir- 
cular motion. 

In this case the radial motion-energy is zero, and the spacial 
separation is constant; no energy-transformation whatever takes 
place and no energy is manifested radially. All points in the 
circle are equally mean energetic conditions. The balance be- 
tween centripetal and centrifugal forces is perfect and perma- 
nent. That is to say, speaking mathematically, 

(centripetal) (centrifugal) 
Radial Force = c M,M,^^ ^^ M + M '"D" ^^^^ 

In the second member of Equation 16 the reader may not imme- 
diately recognize the more familiar expression for centrifugal 



THE MEAN ENERGETIC CONDITION 37 

force, M — The latter, like the other familiar mechanical 
R 

equations which have already been criticised, is a special approxi- 
mation, omitting the sum of the masses ; which is sufficiently 
accurate when one of the masses is so large that variation in the 
other does not appreciably affect their sum. 

If, now, comparison be undertaken between two circular 
motions of the same mass-pair, one at a radius r and velocity v 
and the other at radius R and velocity V, the change in space- 
energy which would be involved in a passage from one to the 
other would be, from Equation 2, 

Ep = cM,M,(^--l-). (18) 

The change in velocity, from v to V, which must occur 
simultaneously in order that centripetal and centrifugal forces 
may remain balanced and the orbit remain circular, will be given 
by Equation 17; or 

v2=(Mi+M,)^ (19) 

and V2 = (Mi + M2)^. (20) 

Therefore the change in kinetic energy involved must be 

But the last term of this equation is one-half of Equation 18, 
the potential energy involved in passing from radius R to 
radius r. The energy absorbed in altering the velocity is there- 
fore one-half the amount, and of opposite sign, from that re- 
leased in increasing the propinquity (if the second radius be 
considered smaller than the first). The net energy released, 
therefore, is the algebraic sum of the two, and is itself equal to 
the last term of Equation 21. In other words, when masses are 
brought into greater propinquity at their mean energetic condi- 
tion, or with circularity conserved, or in permanent equilibrium 
betzveen centrifugal and centripetal forces (for the argument 
applies equally to circular motion or to the mean energetic point 
of elliptical or hyperbolic motion), energy must be abstracted; 
just as it must be when thev are allowed to fall directly together, 
vertically, with no tangential motion. But in the former case the 



38 ENERGY 

amount of energy thus to be abstracted is just one-half of that 
requisite in the latter case. 

The addition to the circular motion of a radial component, 
producing elliptic or hyperbolic orbit, will not affect this universal 
equilibrium between tangential motion and mean energetic 
distance. 

In the case of free, conic-section motion, on the other hand, 
with energy, rather than circularity, conserved, no energy need 
be abstracted at all, as the propinquity increases. Indeed, to 
attain, under a fixed mean energetic distance, a greater pro- 
pinquity at periastron, energy must be added, a.nd in such a way 
that it assumes the radial form. In other words, to produce a 
permanent consolidation of matter, tending to continue in stable 
equilibrium, energy must be abstracted. But to produce a tem- 
porary consolidation, which will reconvert itself promptly into 
separation or disgregation of matter, energy need not be ab- 
stracted. It even needs to be added. 

Further, it is evident from Equation 17 that, as the pro- 
pinquity increases (or the distance of separation decreases) the 
velocity must increase. Therefore, putting Equations 17 and 21 
together, it becomes plain that when masses approach, with 
circularity conserved, energy must he abstracted as the velocity 
increases — which is just the opposite of what is ordinarily held 
to be a universal law. But in the case stated, it is to be remem- 
bered, it is not the radial, but purely the tangential, velocity 
which increases as energy is abstracted. This is one of the many 
ways in which the radial and tangential forms of energy are 
markedly contrasted. 

Thus, for instance, in the case of the moon and the earth, 
which revolve in stable equilibrium and in an orbit which is 
almost circular, a mean energetic distance of about 240,000 miles 
is maintained permanently by a mean linear speed of about 3100 
feet per second. But, if the earth were devoid of rotation rela- 
tively to the sun, the friction of our oceanic tides would be tend- 
ing steadily to slow down the moon. But as this happens the 
decreasing centrifugal force between moon and earth becomes 
no longer able to counterbalance the gravitational attraction be- 
tween the two; and moon and earth tend to fall together. But 
as this occurs, space-energy is released, sufficient not only to 
make good the loss of energy to the tides, but more. The 



THE MEAN ENERGETIC CONDITION 39 

equilibrium remains stable, and the abstraction of tidal energy 
has the apparently paradoxical effect of speeding up the moon. 

As a matter of fact, the earth is revolving, relatively to the 
sun, in the same direction as the moon moves about the earth, 
and with a higher angular velocity. The result is that the fric- 
tional resistance of the tides, instead of tending to abstract 
energy from the moon for the speeding up of the earth, tends 
to slow down the earth's rotation, with the transfer of energy 
to the moon. In this case the moon is simultaneously removed 
from the earth and retarded in tangential linear velocity. 

Either case illustrates the point, viz : that, whereas in radial 
motion the energ}^ increases directly with the linear velocity, in 
tangential motion it increases inversely therewith. The mere 
alteration in direction from radial to tangential, which has been 
accomplished by gravitational reaction with the second mass- 
portion, has constituted a reversal of algebraic sign of the 
energy — algebraic signs, in energetics, being significant merely 
of whether the energy be going into or coming out of a given 
system, or of departure on one side or the other from the mean 
energetic condition. Although it is universally taught, in the 
engineering schools, that energy always enters matter as its 
velocity increases, and vice versa, it now appears that this is 
true of only one-half the energy of the universe — the radial, or 
perceptible, half. With the other half — the tangential, or latent, 
or imperceptible, half — velocities increase as energy departs from 
matter, and vice versa. 

This innate tendency of all vibratory forms of energy period- 
ically to alter the algebraic sign, with every transformation from 
kinetic to potential, or from radial to tangential, or the reverse 
— as visible first in the familiar pendulum, and now again in the 
action of any free mass-pair — is a fundamental characteristic 
of energy which is of the utmost importance. Nature knows 
nothing of smooth, continuous progress. All goes by pendulum- 
swings, reversals and transformations, as a ship tacks in beating 
against the wind. 

It is in such terms as these that the energy-fund of the pair 
must be thought of, as consisting of tangential energy plus 
radial energy. The amount of tangential energy within the pair 
determines the mean energetic distance D, and the tangential 
component of the mean energetic velocity U. It remains perma- 



40 ENERGY 

nently fixed in form and quantity, so long as energy is neither 
added nor abstracted from without. The amount of radial 
energy within the pair determines the eccentricity of orbit and 
the radial component of the mean energetic velocity. It vibrates 
at each revolution from kinetic to potential form and back. 

The Radial Energy-fund. The mathematical expression 
for the fund of radial energy within the pair may be had from 
Equation 7. By algebraic transformations which need not be 
reproduced here, it may be stated in terms of either the mean 
energetic or the periastron condition. It may also be expressed 
in terms of either space or velocity, for the energy alternately 
takes either form. It will also be of convenience for later 
purposes if the kinetic expression be stated in three different 
ways. Thus, 

(potential) (kinetic) 
Radial Energy = E, = 2 c M1M2 -^ = 2 M1M2 — — ^— - 



D ^ 'Mi + M2*l+e^ 



(kinetic) (kinetic) 



^nT^T ^ cos^ a 1 ^ ,T HT U^ sm^ a ,^^, 

= 2 MiM, ^^ , ^^ ^ = 2 e MiMo— ^ri^ — ^ — . (22) 

Of these three kinetic expressions the first and simplest will be 
the most commonly used. Of the others, the last two will some- 
times be convenient because, in the mean energetic condition, 
U cos a is the radial, and U sin a the tangential, component of 
the velocity U. 

Stated in terms of the extreme energetic condition at perias- 
tron, So being the periastron or minimum distance of separation 
and V the periastron or maximum velocity of motion, Equation 
22 becomes 

E, = 2cMA^.^ = 2M.M,^j^-^ (23) 

(It is to be noticed, in order to avoid confusion of thought, that 
at periastron, although the motion is there purely tangential in 
direction, radial energy nevertheless is present; because the tan- 
gential velocity is there so great that the radial forces are un- 
balanced. The radial energy takes this tangential form for an 
instant only.) 

Of all the expressions for the radial energy of a pair, that 



THE MEAN ENERGETIC CONDITION 41 

given in Equation 2, in potential form, is the first to get well in 
mind. Equation 2 reads : 

Ep = cM,M,(-^--i-). (2) 

As the greater distance of separation S becomes very great, its 
influence upon the value of Ep becomes very slight; until, at the 
limit, when S has become of celestial dimensions between bodies 
of earthly magnitude, its value may be neglected. Equation 2 

then becomes proportional to — - only. The same is true of the 

potential forms of Equations 22 and 2^, wherein the space-factor 

appears only as -r^-or-Q-* For this reason it seems convenient 

to assign to this reciprocal of radial space of separation the term 
propinquity; whereupon it may be said that, for all mass-pairs 
not already in unusual propinquity, their potential energy given 
out is proportional to their propinquity. And in any event it 
is true that differences in potential energy are proportional to 
differences in propinquity. 

From this it becomes obvious that the amount of energy 
which must be abstracted from any mass-pair before they can 
be brought into coincidence, with So = o, is infinite. Of course, 
no two bodies can ever be brought into coincidence. But since 
the obstacles to the feat lie only in the solid dimensions and the 
density of the two masses, which are variables to which no rule 
can be applied, it still remains true that the amount of energy in 
a pair which lies awaiting abstraction is indefinite in amount. 
It is only the ability to abstract it which is limited. 

The Tangential Energy-fund. When attempt is made to 
give exact mathematical expression to the tangential fund of 
energy, trouble arises. In tangential motion no force is over- 
come by that motion, as is the case in radial energy. There is 
no transformation of energy. Indeed, there is not even any 
manifestation of energy. When careful thought is taken it 
appears as one of the fundamental principles in nature that 
only radial energy can be perceived by the human senses, or 
by any other external mass-system. A revolving pair, unless 
it be of such dimensions and velocity that its members can be 
perceived separately, as they alternately approach and depart 
from us, does not appear to us as a pair at all, but as two units. 



42 ENERGY 

Thus, the heavenly bodies are all of such dimensions that they 
can be perceived separately, and from them we get our first 
exact ideas concerning tangential energies. Similarly, such re- 
volving pairs as fly-ball governors, fly-wheels, etc., usually re- 
volve at a slow enough speed so that we can perceive their com- 
ponent parts; and so we learn to apply these exact ideas con- 
cerning tangential energies. But if the fly-wheel or the like 
revolves so rapidly, or becomes so minute, that its component 
parts are no longer distinguishable, then we can perceive its 
energy only when we come into contact with it as a whole. We 
then learn — if it does not contribute so much energy to us that 
our wits are at fault — that we have perceived only that portion 
of its tangential energy which has ceased to be tangential and 
become radial. 

While this fact, like so many others which have been adduced 
to the present argument, is of comparatively little importance in 
engineering mechanics, it becomes of basic importance when the 
revolving pair, or system of pairs, reduces to the dimensions of a 
molecule or an atom, and its energy becomes known to us as 
heat or the like. When we touch a body and perceive that it is 
''hot" we are like some vast giant who might be too big to see 
our tiny human contrivances on the surface of the earth, yet 
who might perceive them by placing the tip of his finger upon a 
field in which were many rapidly revolving fly-wheels or buzz- 
saws. He could not see that each fly-wheel or buzz-saw was 
composed of many parts, balanced in their motion in an equilib- 
rium which gives them the appearance of unity ; yet to our senses 
it would be an obvious truth. 

It is therefore not so surprising that, since we have no power 
to perceive tangential energy, we have no means for expressing 
it mathematically. For it is a fact that no exact statement can 
be made as to the tangential energy-fund existent within any 
pair. Tangential energy is there, surely enough ; for it can 
come out, by becoming radial in form, and can overcome re- 
sistance. But there is neither any definite idea nor any exact 
equation for the total amount which can thus come out. We 
know, from Equation 21, that the amount thus available is one- 
half the radial space-energy available from any given same change 
of radial separation. But as the latter has already been shown 
to be, so far as exact mathematical limitations appear, infinite 



THE MEAN ENERGETIC CONDITION 43 

in amount, the only statement which this leads us to is that the 
tangential energy-fund of any mass-pair is one-half of infinity; 
which is, of course, meaningless. It still leaves us forced to 
confess that for the tangential fund we have no exact expression. 

Our knowledge in that direction is much like that, for in- 
stance, of a man who owned excellent thermometers, barometers, 
etc., all of which had slipped their scales. He would know ex- 
actly what a degree of alteration of temperature was like, or an 
inch of barometric pressure. He could report usefully to his 
neighbors, from day to day, upon the changes in the weather. 
But he could never tell them exactly how hot or cold it was, nor 
how far the conditions were from any absolute zero. So as to 
tangential energy, we can measure exactly, in joules, foot-pounds 
or what you please, any stated change in tangential energy, from 
known change of radius of purely tangential motion ; but beyond 
that we cannot go. We cannot determine any absolute zero or 
maximum from which to make our ideas exact.* 

In thinking of the energy-fund of any mass-pair or system, 
therfore, all idea of reducing the thing to any exact statement, 
founded upon an absolute zero, must be abandoned from the 
start. The only base or zero which has become visible in the 
preceding discussion — which has been just as exact as it has 
been possible to make it, short of dealing with infinitesimal 
masses — has been the mean energetic condition. It is only in 
this mean energetic condition that the tangential energy is directly 
visible. It is the amount of tangential motion on hand which 
determines the mean energetic distance D, or the space occupied 
by the pair. 



*In addition to the above it must be noted, before leaving this contrast 
between radial and tangential energies, that the radial energy alone may 
be considered the exclusive property of the mass-pair itself. It has already 
been noted that between a pair of absolutely single bodies only radial 
separations and velocities may be measured. For tangential motions the 
presence of at least three mass-portions is necessary. In nature this is 
tantamount to saying that no body is truly a single body. Every natural 
body possesses dimensions ; and if so, tangential motion of revolution can 
be perceived by comparing its various portions. It is thus that we per- 
ceive the motion of the moon around the earth ; as different portions of 
the earth swing up or down relatively to the moon, we say that the moon 
rises or sets. But to those telescopes which may be situated upon planets 
so distant that the earth appears as a point of light, the revolution of the 
moon can become perceptible only by a comparison with distant and 
apparently fixed stars; which amounts to bringing in a third mass very 
much larger than any of those yet mentioned. 



44 ENERGY 

On either side of this mean energetic condition the radial 
energy of every energetic pair vibrates as does a pendulum about 
its supports. But, in the case of the free energy-system, this 
support, instead of being fixed and definite, is floating in mid- 
space. It cannot be attached to or measured from any basis 
which has yet been devised. The most which can be done is to 
compare it with other mean energetic systems, each of which is 
equally homeless. As the sun serves as a central basis in refer- 
ence to which the motion of the planets may be conveniently 
measured, so the mean energetic condintion serves as a central 
basis or zero-point from which, in either direction, the radial 
energy of the system may be conveniently measured. But both 
the sun and the mean energetic condition float indeterminately 
in space, without any possible reference to any absolute base or 
zero. 

Indeed, the prime lesson sought to be imparted by this paper 
is that there is not anywhere in energetics, in any department 
where exact knowledge has yet penetrated, any reason for ever 
believing in the existence of an absolute zero for anything. So 
far as we now know, there is no absolute zero for either energy, 
velocity, space, force, volume, temperature or entropy; nor, so 
far as the writer is aware, for the similar factors of those other 
forms of energy with which he is less familiar, such as chemical, 
electrical, etc. All that we have ever perceived in any of them 
is a vibration on either side of some central mean value, which 
itself, in turn, cannot be regarded as fixed. 

While this topic must receive further analysis, in later papers 
of the series, before it can be understood, it should be an attribute 
of energy familiar to every student, as its basic characteristic, 
that energetic potentialities consist as much in departures upon 
one side of the central mean as the other. For instance, in 
thinking of the energy embodied in mundane gravitation, such as 
water-power, it should be remembered that very low conditions 
of matter embody energy, as well as very high ones. When 
one wishes to purchase water-power it is usual to buy some 
basin which nature keeps filled with water at an appreciable 
elevation above the sea-level — which latter Is In this case the 
mean energetic level. But If the valley of the Salton Sea, In 
California, or the Caspian or Saharan basins, or any other simllai 
depressions below the sea-level, which nature would keep emptied 



THE MEAN ENERGETIC CONDITION 45 

of water by evaporation, happened to be conveniently near the 
sea and on the market, they would constitute invaluable water- 
powers. With their help the waters of the ocean, which are 
commonly regarded as having reached the absolute zero of 
hydraulic head, would become gigantic sources of hydraulic 
energy. 

Similarly with heat-engines, it is only by chance that it 
happens to be the most convenient thing, when one wishes 
power, to buy coal, fit to give out some 15,000 B.t.u. per pound 
at a temperature some hundreds of degrees above the mean 
thermal level for the surface of the earth. On the other hand, 
if a mine should be discovered from which could be procured a 
durable solid which would bear transportation, and which would 
absorb, instead of develop, some 15,000 B.t.u. per pound at a 
temperature even two hundred degrees below that thermal mean 
(whereas ice will absorb only about 140 B.t.u. per pound at a 
temperature of 32° Fahr.), the owner could sell it in unlimited 
quantities for heat-engine purposes, for chilling the condensers 
so low that ordinary sun-heat, even in winter, would suffice for 
the motive heat. If this novel substance happened to be more 
convenient than coal for any reason, all our better heat-engines 
would soon be designed in every line with a view to its use, just 
as they now are for the use of coal-made steam, or gas, or oil. 

Nor is this mere phantasy. Since the temperature of our 
condensers is just as far below that of our steam-boilers as that 
of the boilers is above the condensers, our steam-engines must 
be regarded as evincing the availability of cold for work-per- 
formance as much as that of heat for the same purpose. If any 
steam-engine owner does not beheve that he is running a cold- 
engine, let him allow his condenser to warm up. He will be in 
exactly the same trouble as if he allows his boiler to cool oflF. 
And as a matter of fact it is quite as likely that there exist 
somewhere in the universe deposits of substance which embody 
chemically the chill of interstellar space as that here on earth 
are deposits of coal which embody chemically the incandescence 
of intrasolar mass — though they are not likely to be convenient 
solids, like coal. 

The conclusions which have been sought in the foregoing 
argument may be summarized briefly as follows : 



46 ENERGY 

1. The energy-fund of any mass-pair is made up of two 
contrasted sorts, viz : the radial and the tangential. 

2. Of these, only the radial fund is capable of exact mathe- 
matical expression ; or, so far as the argument has yet proceeded, 
of manifestation to any external mass-system. 

3. The variations of either fund occur, not toward or away 
from any known absolute zero of any condition, but on either 
side of a mean energetic condition; and this mean energetic 
condition itself cannot be located absolutely, but only relatively 
to other mean energetic conditions. 

4. The tangential energy-fund, itself unmeasurable, acts as 
mean central base for the radial fund, on either side of which the 
latter vibrates. For the tangential fund the mean central base 
as yet remains undefined; although it is obvious, from the 
generally stable equilibrium of energetic phenomena, that such a 
stable central condition must exist. 

5. Of the total energy-fund of any mass-pair, the radial 
energy constitutes the perceptible, the measurable and the trans- 
missible portion. It is the medium of communication between 
the pair and the outside universe. It may properly be styled the 
sensible energy. In its potential form it is distinguishable as 
either an unusual degree of spacial separation, or as an unusual 
lack of force (as at the apastron of the illustrative elliptic 
orbit). In its kinetic form it is distinguishable as either an 
unusual degree of unbalanced force, or as an unusual lack of 
spacial separation or unusual concentration of mass (as at 
periastron of the illustrative ellipse). These statements will be 
found to apply broadly to the thermal and mechanical energies 
of all forms of matter, as well as to the elementary illustration. 

6. Of the total energy-fund of any mass-pair, the tangential 
energy is the imperceptible, the immeasurable and the non- 
transmissible portion. It is the elastic base upon which rests 
the radial energy. It is the means for the storage of perceptible 
energy received radially from without, and likewise the source 
from which is drawn the energy manifested radially to the out- 
side world. It may properly be styled the latent or invisible 
energy. It is not to be perceived directly at all ; and indirectly 
it is to be distinguished solely by its ability to turn into or 
absorb radial energy. 

7. All energetic measurements must be made from, and all 



THE MEAN ENERGETIC CONDITION 47 

concepts based upon, not any absolute zero of anything, which 
exists as a fixed support, but from a central, mean-energetic 
condition, which itself floats unsupported in mid-space, like our 
sun in the heavens. While the position of this mean-energetic 
condition must itself be subject to natural law, yet it is con- 
trolled by forces and phenomena too large to be taken into 
consideration in any concrete case. 

8. Every mass-pair embodies some radial energy and some 
fund of tangential energy. This fact has not yet been developed 
in the argument, but it should be stated here, in company with 
the preceding seven principles. When the eccentricity of the 
mutual orbit of the pair is great, the radial fund is large in 
proportion to the tangential. When the eccentricity is small, the 
radial fund is small. But as the eccentricity can never be con- 
ceived as becoming either zero or infinity, there must always be 
some finite fund of radial energy. And since the pair can never 
be conceived as united into complete coincidence, but must always 
occupy some space, there must always be some fund of tangential 
energy. This idea will be developed later. 



CHAPTER IV. 

The Two Factors or Dimensions of Energy. 

In all of the. mathematical expressions for energy which have 
been developed in the preceding chapters there have everywhere 
appeared two factors. One of the factors is the product M^Mg 
of the separate masses involved. The other is some function of 
either the space or the motion involved in their separation into 
duality. It is obvious that these two factors possess a distinct 
and contrasted significance. 

The first of these factors, M^Mg, is a measure of the extent 
to which duality exists, in which may be embodied the space or 
motion relationships which are measured by the other factor. It 
is a measure of the extent of "mass-pairing," as we shall call it 
for convenience, which is present. It has therefore been named 
by the writer the '"extent" of energy, or the ''extensity," present. 

The second factor gives the degree of space or motion em- 
bodied within the mass-pair; and as it appears to human senses 
as the feature of energy which gives evidence of the degree of 
concentration of energy, by the sharpness of sensation or other 
effect which the energy may produce, it has been called by the 
writer the "intensity" of energy present. Of the two factors 
this latter is the more familiar to students of energetic phe- 
nomena, and will therefore be considered first. 

The Intensity of Energy. If the fundamental expressions 
for potential and kinetic energy be divided by the factor MJA.^, 
so as to derive mathematical expressions for the intensities of 
these two energies, there results, for the potential intensity, 
from Equations 2 and 22, 

Ip = c(^-^)=-g|-(S-S„) = 2c^ (24) 

and for the kinetic intensity, from Equations 5 and 22, 

1 V^-V 2 U^ e 

^ 2'Mi + M2 Mi + M2*l+e2 ^^ 

From these equations the Intensity of energy appears broadly as 

48 



THE TWO FACTORS 49 

a function of (i) for potential intensity a difference in ''pro- 
pinquities/' or reciprocals of spacial separation; and (2) for 
kinetic intensity a similar difference in ratio of velocities-squared 
to aggregate mass involved. 

In kinetic mechanical engineering, it is the prime charac- 
teristic of the intensity of energy that it controls the direction 
of energy-transformations. It is a fact familiar to engineers 
that it is the body possessing the higher velocity, even if the 
smaller mass, which overtakes, collides with and gives its energy 
to the more slowly moving masses. Our entire experience with 
hammers, projectiles, and railway-collisions is but experience 
with the intensity of kinetic energy, and its promotion of energy 
transformation. 

But now it appears that, whereas our present sub-conscious 
idea of kinetic intensity attaches itself merely to velocity, the 
exact affair is in reality the ratio of velocity-squared to aggregate 
mass. Like all other energetic phenomena, its manifestation 
here upon the earth's surface, with the earth for one of the 
participating masses in every pair, has been disguised by the 
fact that here the aggregate mass, viz: projectile plus earth, 
remains virtually constant, and so has been dropped from con- 
sideration of the variables. But when the bodies concerned are 
of molecular magnitudes, with nothing known as to comparative 
masses of the members of any pair, it is obvious that the de- 
nominator of the ratio mav become as active a factor as the 
numerator, in the variation of intensity. 

As to the potential intensity of energy, that is also a familiar 
fact in every-day engineering; but not quite in the form pre- 
sented here. 

In the first place, potential intensity, or intensity of energy 
due to an unusual departure of spacial separation from the 
average, may assume either of two forms, viz : unusual separa- 
tion, on the one hand, or unusual lack of separation on the other. 
That is to say, potential energy may be observed as great either 
because S is very much greater than the mean energetic distance 
D, or because So is very much smaller. 

All of the cases in engineering w^hich are commonly recog- 
nized as potential mechanical energ}^ such as suspended weights, 
mill-pond water, or projectiles at the summit of their trajectories, 
belong in the former class. They are in reality following the 



50 ENERGY 

apastrons of extremely eccentric orbits, which pass closely about 
the center of the earth — which orbits, of course, can never be 
completed because of the solid obstruction of the earth's body. 
Their mean energetic distances are a few miles at most, perhaps 
a few feet only, from the earth's center; whereas their separa- 
tions from that center while we are using them is some four 
thousand miles, the earth's radius. Moreover, they possess very 
little tangential motion parallel with the earth's surface. There- 
fore, their gravitational force is almost entirely unbalanced. We 

• c • • 

consider —— , m Equation 24, to be a constant, and So in the 

parenthesis to be negligible. The distance of separation S — or 
h, for height or head, as it is commonly called — remains alone, 
ready for us to consume of it what small fraction we may. 

But we also know potential intensity, in engineering, as an 
unusual lack of separation in mass. As we compress an elastic 
gas, or a steel spring, for instance, into a volume less than its 
condition of mean energetic equilibrium, there is developed a 
latent potentiality for energy which manifests itself as an unusual 
intensity of force. This force does not appear to the average 
engineer, it is true, as an integration of the many unbalanced 
centrifugal forces of a multitude of tiny orbits at periastron. 
But there is no other unbalanced force in the elements of 
mechanics which so fits the conditions as to be acceptable as an 
explanation of it. 

I X I 



Force, it is also true, is a function not only of — - (or - 



S volume 

as we should state it for gases), but also of the masses involved. 
Therefore it is not a pure manifestation of intensity. But when 
the mass-factor happens to be constant, as is commonly true in 
engineering mechanics and as may sometimes be true even in 
molecular mechanics, it becomes a true measure of intensity. 

In this case, also, the characteristic of intensity as the de- 
terminator of the direction of energy-transformation appears. 
It is always the greater force which overcomes and contributes 
energy to the smaller force. It is always the smaller force which 
receives and stores it. That is why our fundamental concept of 
energy covers the action of all the unbalanced force present, 
acting through an Infinitesimal distance, rather than all the dis- 
tance covered, acted upon by an infinitesimal element of the force. 



THE TWO FACTORS 51 

That is to say, the mathematical equation for this fundamental 
concept of an infinitesimal element of energy is Force X 
d {Space), and not Space X d (Force). 

Therefore, while the mathematical equations are the only true 
guides, yet the general concept of potential intensity may be based 
upon either force or space, force being understood to be synony- 
mous with lack of space, or reciprocal of space. When the in- 
tensity is in the form of unusual space between the mass- 
portions, the unbalanced force is centripetal, gravitational or 
concentrative, and is comparatively slight. When the intensity 
is in the form of unusual lack of space, the unbalanced force is 
centrifugal, or disgregative, and is very great. The first offers 
us much distance traversible with little force; the latter makes 
available a greater force operative through a smaller distance. 
The first typifies the action of suspended weights ; the latter that 
of compressed elastic matter. 

Thus, either unusual attenuation or unusual condensation of 
matter constitutes potential energy. Unfortunately, at present 
we quite lack suitable names for these two contrasted types of 
potential energy. If we were to coin words for them, disgregic 
and congregic energies would be the natural names. But this is 
merely a suggestion. 

In the above statements all words of degree, such as great 
or little, are used only in a comparative sense, for any given 
mass-system. For in natural mechanics all human standards of 
dimension disappear. The mass-action of solar systems, base- 
balls and molecules is now believed to be the same, in principle. 
Any such a system may be taken as a standard of dimension; 
but the ideas of great or small applied from these different 
bases must be understood as fitting widely different experiences. 

The Extensity of Energy, or Degree of Mass-pairing. 
The extensity-factor of energy M^Mg appears always as a 
product of two separate masses, rather than as their sum. This 
is so because energy can exist only betzveen masses, and not 
throughout mass. That is to say, the symbols M^ and M^ signify 
that within the first body there reside M^ units of mass, while in 
the other body reside M2 units. Any single mass-unit in M^ 
would then form, with the several single units in the other body, 
and across the gap between the two bodies, M, unit mass-pairs. 
The entire community of mass-units in M^ would therefore form 



52 ' ENERGY 

Ml X M2 unit mass-pairs across the gap separating the two 
bodies. 

Remembering that energy can exist only in the gap between 
separate mass-portions, and never in the mass itself, it becomes 
clear that the factor M^Mg is the extent of mass-pairing which 
is energetically active across this gap. It is clear, too, that in 
any energetic system the total extent of mass-pairing present, or 
2 MM, in which to embody intensities of relationship, is just as 
much a factor, and may be just as variable a factor, in the 
energy as is the intensity of relative space or motion itself. It 
is because we have so long been accustomed, in our engineering, 
to splitting off from the earth the mass of a railroad-train or a 
cannon-ball and treating only of its energy relatively to the earth 
— in which case the mass-pairing factor remains constant so 
long as the separation between earth and machine, or the energy 
itself, lasts — that it has been forgotten that the mass-pairing 
factor is itself as hkely to experience variation as is the intensity- 
factor of relative space or motion. And in molecular mechanics 
it is perhaps even more likely. 

This variation in extent of mass-pairing may take place in 
either of two ways. First, the mass of -either party to the energy 
may increase or diminish ; in which case Mj^ -j- M„, or S M, 
varies with 21 MM. This, according to the ordinary teachings 
of mechanics, is what always occurs. We are taught, in fact, 
that we can increase the energy between cannon-ball and cannon 
(for a fixed muzzle-velocity) only by increasing the mass of 
both cannon and projectile. 

But there is a second method of increasing the massivity of 
energy present, and that is by increasing the number of cannon 
and projectiles. At first glance, this seems to be the same as 
before, viz : energy increasing in proportion with mass. The 
difference appears when it is remembered that the earth is a 
party to almost all engineering energies, and that their true 
nature comes out only when matters are expanded to a scale 
commensurate with the dimensions of the earth. The earth 
weighs about 6 X ^^^ long tons. A cannon and projectile 
which together weighed this amount, and yet possessed only the 
muzzle-velocity of standard cannon, would be a very mild affair, 
as planetary energies go. But compare with this 6 X 10^^ long 
tons of standard cannons and projectiles, all going at once, and 



THE TWO FACTORS 53 

each of the projectiles equipped with energy relatively to the 
entire remaining mass — and some concept may be had of the 
energetic possibilities of mass-subdivision! In comparing these 
two cases, the 2 M, or aggregate mass, of the two are the same ; 
but in the latter case the 2 MM, or extent of mass-pairing, is 
very much the greater. 

The distinction involved here is hard to bring out clearly, for 
man has had no experience with mass-systems consisting of an 
earth's-weight of cannon, all going off at once. Yet this is the 
only true mechanical simile, in comparison with the placid old 
moon-earth system, to a white-hot cannon in comparison with 
the same cannon cold and solid, discharging a single shot. It is 
because human experience covers no gradations between these 
two extremes, of matter acting as a solid unit on the one hand 
or as an innumerable multitude of separate solid units on the 
other, that we have always regarded "work" as one thing and 
"heat" as a very different thing— that we have arranged all our 
formulae for mechanical energy with the mass-factor appearing 
as M instead of as M^, neglecting entirely the wide range of 
variation of which the factor 2 MM is capable while S M 
remains constant. 

Yet this distinction is all-important, as lying at the heart of 
the comprehension of any of the more obscure forms of energy, 
such as heat. In order to grasp it, it must be remembered how 
necessary has been the common habit of speaking, even in 
celestial mechanics, of the bodies involved as if they were solid 
homogeneous units. In discussing the energy embodied between 
moon and earth, for instance, it has been customary to consider 
both moon and earth as perfectly solid, unit spheres, each acting 
as if concentrated at its own center. Yet this is very far from 
the truth. Each of these "units" is in reality extensively and 
minutely subdivided into interacting energetic mass-pairs. We 
who inhabit the earth know full well of that planet, if not also 
of the moon, that every energy which forms the subject of a 
human science, in mechanics, hydraulics, meteorology, chemistry, 
biology and electricity, stands in evidence of minute subdivision 
of mass. The earth, instead of being a homogeneous unit, is in 
reality subdivided into a vast and most intricately organized host 
of parts, of differing densities and other characteristics ; and 
between these divided, distinct and independent particles, of 



54 ENERGY 

rock, water, air and the various chemical elements, exist the 
most diverse funds of energy, both potential and kinetic. Merely 
to mention the unquestionably mechanical ones, there are the 
energies of the v^inds, v^aves, tides and waterfalls, all the pon- 
derous machines of human construction, and the innumerable 
host of moving arms, legs, wings, fins and claws besides. There 
is not only energy in the abysmal gap between moon and earth, 
but there is energy in each of the myriad of tiny crevices in each. 
Wherever exists subdivision there exists energy. 

Indeed, all that is rneant, in speaking of the moon and earth 
or any other body as a ''solid" unit of mass, is that for the par- 
ticular purposes of this particular sort of energy the mass- 
portions concerned may be considered as solid units ; that is, 
acting as if their mass were concentrated at a single point. So 
long as this special assumption holds good, 2 MM must be 
regarded as a constant, as well as S M, and the amount of this 
particular sort of energy is proportional to the first power of 
the mass involved. But, considering all sorts of energy together, 
we know of no instance in which this special assumption holds 
true broadly. Human experience has never yet encountered a 
homogeneous or true solid. Like the geometric point or line, 
the solid state of matter is merely a convenient figment of the 
human brain. 

It has been altogether natural, and even necessary, in the 
study of applied mechanics, to consider mass thus, as lumped 
into portions which were solid units, acting perfectly in unison. 
But for a mechanical concept of heat such an idea is fatal; and 
even for the purposes of applied mechanics it seems to the 
writer to have been overdone. For it is essential, for the com- 
prehension of any of the several other energies which are daily 
growing in importance, to have constantly in the mind's eye a 
picture of mass, not as a homogeneous unit, or solid, but as a 
heterogeneous system, subdivisible again and again, into smaller 
and smaller portions, just as far as human perception is able 
to penetrate — into molecules, atoms, ions and what not. The 
history of science justifies no other view. Each new stage of 
scientific progress has revealed some further refinement of sub- 
division of matter which had previously escaped the more clumsy 
perceptions of cruder times. 

The difference to the comprehension of the general nature of 



THE TWO FACTORS 55 

energy lies in the fact that if the homogeneous dogma be true, 
the purely mechanical energy which may be embodied in any 
given mass is limited by the velocity or distance relatively to 
some fixed base, such as the sun, v^hich may be imparted to it; 
in short, to a variation in the intensity-factor of its energy. 
Thus, in comparing the energetic possibilities of different masses, 
these must vary directly as the mass. But if the heterogeneous 
dogma, based upon the equations given in these papers, be prop- 
erly understood, then mass appears as capable of embodying 
within itself any amount of energy whatever, not only by varia- 
tions in its intensity of energy, but also by variations in its 
extensity of subdivision, or comminution. 

This can perhaps be made clearer by an illustration. 

Let us consider a form of energy which we may call military 
or combative energy. Suppose there exists an army of twenty 
thousand men. If one of these men should desert his duty, the 
power of the remaining nineteen thousand and odd arrests him 
— perhaps only after some desperately violent "intensity" of 
resistance, the utmost combative energy that any one man might 
arouse in the face of superior numbers. This, in the particular 
illustration chosen, is the minimum possible degree of sub- 
division of the army's mass into combatively opposed portions. 
The extensity of combative energy present is given by multiplying 
one by 19,999 ; while the intensity of combative energy depends 
upon the strength and spirit of the individual thus split off from 
the corps. 

The portion of the army thus split off, in disobedience, of 
course might be greater than a single man. The number in 
revolt might grow most gradually, man by man, until it had 
become full half the army. If we suppose, for simphcity, that 
each man's intensity of combative ability were equal to that of 
his fellows, then the energy of combat would grow, as the 
revolt spread, proportionately to the product of the numbers 
engaged upon the two sides. And this would reach a maximum, 
at 10,000 X 10,000 = 100,000,000, when the opposed parties had 
become equal. 

But this is not the maximum extensity of combative energy 
of which the army is capable. It is merely the maximum attain- 
able when only a single subdivision, into only two portions, 
occurs. Suppose, however, that by the time the revolt had 



56 ENERGY 

absorbed one-half of the army it became civil war, one side 
adopting a red uniform and the other a blue. Suppose, then, 
that the spirit of disaffection so spread that further subdivision 
took place within each faction. The red army splits into a 
"yellow" and a "gray," of five thousand men each, while the 
blue army splits similarly into a ''green" and a "purple." Each 
of these new armies raises a banner of its own, and opposes any 
and all comers. What is now the possible number of individual 
personal conflicts? What is now the extensity of military or 
combative energy? 

Between the four armies are possible six different battle- 
arrays — a yellow-green, a yellow-gray, a yellow-purple, a gray- 
green, a gray-purple and a green-purple. But the possible num- 
ber of personal conflicts in each battle has now been reduced to 
only 5000X5000 = 25,000,000, or one-quarter as great as 
before. The aggregate number for all the armies, together, there- 
fore, is but 6 X 25,000,000 = 150,000,000. This measure of the 
extensity of combative energy proves to be only fifty per cent, 
greater than when the army was divided into only two equal 
portions. 

But, obviously, the extent to which the army may subdivide 
into equal portions, in mutual discord, is not limited to four such 
parts. The possibilities in this direction are not exhausted until 
each individual soldier has become a knight errant, under his 
own standard, and the field a proverbial Donnybrook Fair. The 
extensity of combative energy would then be much greater than 
150,000,000, it is clear; and yet it is equally clear that it would 
not have grown in proportion to the number of subdivisions. 

Also, it is to be noted before passing to mathematical exact- 
ness in the discussion, the degree of subdivision of the army is 
limited to the degrees specified, only because the discussion has 
been confined to military energy, a form in which the unit mass- 
factor is a single soldier-pair. That is to say, for this special 
purpose, it has been assumed that each soldier is a solid, homo- 
geneous, indivisible unit, containing no internal subdivisions and 
energies. This, of course, is not true In speaking of energies in 
general ; for each soldier is a most intricate conglomeration of 
separate organs, muscles, glands, bones, cells, etc., and is capable 
of embodying much physiological energy even in times of peace, 
when the army remains a solid unit. Only, for the particular 



THE TWO FACTORS 57 

purposes of combative energy the unit mass-pair is a couple of 
soldiers, completely equipped ; just as for the particular purposes 
of heat-phenomena the unit mass-pair must be considered as two 
somethings called molecules, for chemical energy two somethings 
called atoms, for electrical energy two somethings called elec- 
trons, etc., etc. 

If this general idea be reduced to a mathematical basis, it 
will appear that the extent of mass-pairing, or extensity of 
energy, X, of which any aggregate mass M is capable, grows 
with the number of equal parts n into which the aggregation is 
subdivided, according to the equation 

X = 4MM1-^) (26) 

From this, if n=i, X=o. If n=2, X^iM^. If n=:4, X = 

-A_M-. If ni=ioo, X=:o.495M^ And as n grows indefinitely 

larger the value of X approaches more and more nearly and 
slowly to JM-, which value it can never reach. 

On the other hand, if n becomes less than two the value of 
X becomes a fraction of ^M-. Such would be the case when the 
aggregate mass was divided into two portions, but not equal por- 
tions, the value of n becoming one plus a smaller and smaller 
fraction as the one portion became a smaller and smaller fraction 
of the whole. Such is the case, for instance, in the energetics 
of engineering, if the mass of the earth be considered as the 
unit of measurement. As we split off from the earth portions 
which we manufacture into cannon-balls, railroad-trains, etc., 
we are in reality giving to n of Equation 26 the value of one 
plus a very small fraction. The energy becomes zero, because 
its mass-factor has become zero, only when the value of n 
becomes unity, signifying that the earth is a homogeneous solid, 
possessing no dissevered parts like cannon-balls. 

If n becomes less than unity, the value of X becomes nega- 
tive. Such would be the case when the entire mass in question 
is but a part of a single member of some much larger mass-pair. 
As the value of n, which must always be a positive quantity, 
approaches zero the value of X approaches negative infinity. 

The application of this formula to the understanding of 
energetic action will be made in later papers. 



58 ENERGY 

Hitherto the principles of mechanics have usually been dis- 
cussed in the text-books only in terms of constant mass-quan- 
tities. Indeed, there is no college-treatise yet come to the writer's 
attention which gives to the student even an inkling of the fact 
that the mass-factor in energy may be — to say nothing of the 
fact that it always is — just as variable and active a factor as the 
space or motion factor. For the purposes of applied mechanics 
this is quite sufficient; but so soon as the general field of ener- 
getics is entered — as it must be by every modern student in 
even the specialized branches of engineering, wherein are con- 
stantly met transformations between heat, work, chemical and 
electrical energies — the method promptly becomes disqualified. 
The student should be taught, as soon as any general energetic 
concepts whatever are presented, which is usually in the study of 
thermodynamics, in the junior year, that the mass-factor of 
energy is just as frequently and widely a variable as is any 
other; and that the mass-factor is not merely mass, but may be 
anything between the first and second powers of mass. 

To accomplish this, between the course in the elements of 
applied mechanics (usually taught as a part of the general course 
in physics) and the course in thermodynamics, should be in- 
serted a separate course upon the true elements of mechanics and 
mechanical energetics, as outlined above. The approximate for- 
mulae of applied mechanics the student is to continue to use in 
his engineering problems. The true formulae are to furnish 
him with his concepts of truly natural mechanical action, with 
which he is to interpret the obscure phenomena of thermal, 
chemical and electrical interactions. 

In this, the basic fact now needing especial emphasis is that 
when energy is imparted to mass it may find embodiment therein 
in either of two general ways; and in nature these two ways 
occur, not only with equal frequency, but always in combination. 
First, It may Increase the relative velocity of motion, or space 
of separation, of mass-portions which are already in separate 
existence ; that Is, it may Increase the intensity of energy. Sec- 
ondly, it may subdivide mass which was previously unified into 
newly separated mass-pairs, Into each of which Is injected a first 
measure of relative separation or relative motion, or both; that 
is, it may increase the extensity of energy. 

It has been too commonly taught that only when we raise 



THE TWO FACTORS 59 

weights or accelerate cannon-balls do we embody energy in mass, 
and too seldom taught that when we crush rock or grind cement 
we do the same. Only, when the cement is ground or the rock 
crushed the resultant particles do not embody, in their disgrega- 
tion, an energy which we can utilize again. So that we say that 
it is "gone" ; although all of it which has not become heat lies 
there right before our eyes. And even as to the heat, that 
appears, upon inspection, to be merely a finer degree of comminu- 
tion and disgregation than that embodied in the visible particles. 
For not only will the same processes which grind rock into 
powder also grind water into steam-heat, but the latter is, almost 
equally with the former, unavailable for further energetic use. 

The student has been taught from the start, in the doctrine 
of the Conservation of Mass, that mass can be neither created 
nor destroyed. But has he been taught with equal care that 
there is no known limit to its aggregation or subdivision? Or 
that these processes are going on at all times, in nature? Or 
that energy is as much involved in these processes as it is in the 
accumulation or dissipation of motion or space? 

Nor can any limit be placed to the value to the student of an 
exact concept of this great natural fact, in after life. It may be 
only suggested here how wide may be its useful application. It 
is not alone in thermal, chemical and kindred phenomena that 
subdivision, specialization and organization of subject-matter into 
cooperation are of prime importance for the effectiveness of 
energy. Before the student has reached college he has learned, 
upon the play-ground, to substitute skill for bull-strength ; he has 
learned that athletic energy must be subdivided and organized 
into team-work before games can be won, or even played, which 
are worth while. As college-student and embryo manufacturer 
he learns that shop-organization and office-specialization are 
prime factors of success in business. If he joins the militia he 
learns the importance, in military or combative energy, of the 
solidification of bodies of m.en for the resistance of shock; but 
of their subdivision and specialization to a high degree, until 
virtually each individual does a different thing, in order to 
embody in them the highest degree of combative effectiveness. 
Should he study law he learns from the most eminent jurists 
that the whole business of the law is to define and determine the 
relationships which are to prevail, as natural, between the 



60 ENERGY 

individuals of our ever more numerous, more diversely intricate 
and more energetic race; and natural relationships are just what 
these pages are to define and make familiar, by study of them 
as they exist between the simplest possible individual elements. 

It is further the writers' prediction that before to-day's 
student has reached middle-age he will have learned, from 
forcible and costly experience in helping to make his country's 
history, that what that country has needed most, for decades, is 
a similarly accurate concept of the natural relationships between 
man and man, in the day's work. He will have learned that 
what it has long urgently needed, and soon can exist no longer 
without, is an economic organization, throughout its every eco- 
nomic activity, similar to that now prevailing within each factory. 
Within each factory we now have superlative discipline; but 
between our factories prevails superlative anarchy and civil 
strife. We are relying too much at present, for prospective cure 
of our economic troubles, upon greater intensity of individual 
effort. We are too unconscious of the possibilities of greater 
extensity of coordinated energy. We scarcely know yet what 
the words mean. 

Whether a student is to become an engineer, a teacher, a 
business-man, a lawyer, a preacher or a statesman, his life-work 
should be founded upon a clear understanding of mechanical 
energetics. In each of those fields, if his education is to aid his 
life-work, he miust have at his fingers' ends the fact that energy 
can be accumulated by accumulating mere mass ; but that then it 
partakes of the nature of solid mass. Solidity, rigidity, 
inflexibility, hardness are its characteristics ; impact and friction 
are its results. But that if, with the accumulation of mass — or 
even v/ithout it — go subdivision, specialization and coordination 
into increasing extensity of mass-pairing in energetic interaction, 
then fluidity, flexibility and efliciency in work-performance be- 
come its characteristics. In addition, the amount of energy 
which can then be embodied in any given mass is extended indefi- 
nitely, absolutely without limit foreshadowed by those mathe- 
matical formulae which are yet available. This is as true of 
masses of men as of masses of matter. 



CHAPTER V. 

The Extreme or Critical Energetic Conditions. 

In considering the mechanical elements of the original freely 
moving mass-pair, early in this series of papers, it was established 
that its energy vibrated, during its progress along a conic-section 
orbit, on either side of a niean energetic condition. This mean 
energetic condition was identified as occurring at the extremities 
of the latus rectum of the conic-section orbit, twice in each 
revolution. 

The aspects of the different possible forms of orbit which 
were listed in Chapter II, viewed in reference to this mean 
energetic condition, may be stated as follows : 

1. If the orbit be circular all points of the orbit embody the 
mean energetic condition, and no energy-transformation what- 
ever occurs. The pair is perceptible only as a unit. Its orbital 
motion is evident to man only in its occupancy of space and its 
resistance to compression; that is, by its permanence and in- 
destructibility. 

2. If the orbit be elliptic there are two mean energetic points 
in each revolution, through which energy-transformation occurs 
periodically, first in one direction and then in the other, like the 
swing of a pendulum ; and this process continues indefinitely. 

3. If the orbit be parabolic, while there are still two mean 
energetic points per revolution, yet there can be only one swing 
of the pendulum ; and on its outward swing the energy-trans- 
formation tends toward a complete balance, with no residue. 
That is to say, as the bodies swing apart, converting kinetic 
energy into potential, the former is just almost, but not quite, 
absorbed in accomplishing the utmost separation which is imag- 
inable for the pair. As the bodies separate very widely their 
velocity Slmost becomes zero ; but it never quite does so, for the 
gravitational attraction will then also have become almost zero, 
so that there is little tendencv to stop separating and reverse into 
a mutual approach. But this perfect balance of kinetic and 
potential energies may be regarded as so very unlikely as to be 

61 



62 ENERGY 

non-existent in nature. Virtually, all orbits are either elliptic or 
hyperbolic in form and nature. 

4. If the orbit be hyperbolic there is not merely only one 
swing of the pendulum and only two mean energetic positions, 
but there occurs permanent dissociation of the pair afterwards. 
The kinetic energy at periastron is so great that, even after 
separation to an infinite distance, there still remains a finite 
residue of velocity of separation, which can never be absorbed 
potentially by the pair. 

5. If the orbit be a straight line — which is the limiting case 
of an hyperbola with infinite eccentricity — there is no discernible 
mean energetic condition, no appreciable perturbation of path 
by the pair's propinquity (which is zero), and no exact measure 
for the energy involved. All quantities have passed to either 
zeros or infinities, neither of which the mind of man can grasp. 
The situation is to be mentioned, not as a natural fact, but as a 
mathematical limiting condition, to which natural phenomena 
may never attain ; and also to remind the student how un- 
natural is the straight-line path of motion, followed by matter 
only when under constant constraint. 

Let this question be illustrated by means of a somewhat 
familiar mundane mechanism. 

Imagine a cannon placed upon a platform elevated ten miles 
above the surface of the earth, aimed horizontally, and fired. Its 
projectile will start upon an orbit which is commonly described 
as a vertical inverted parabola. But this is true only upon the 
erroneous assumption that the earth is a flat solid of indefinite 
horizontal extent, its lines of gravitational attraction being par- 
allel vertical ones, of equal intensity at all distances from the 
earth. But Newton himself proved that the gravitational efifect 
of a sphere of appreciable dimensions followed his law exactly, 
even when distances very small in proportion to its diameter 
were considered, just as if the mass were concentrated at the 
sphere's center. The true definition of the projectile's path is 
therefore an ellipse, having its further focus coincident with the 
earth's center. This path, however, the projectile cannot prose- 
cute far before it is interrupted by collision with the earth's 
solid surface. 

Suppose, however, that muzzle-velocities might be increased 
indefinitely. The process would in reality be one of adding 



CRITICAL ENERGETIC CONDITIONS 63 

energy to the pair tangentially, at apastron. The ellipse would 
then grow wider, the eccentricity less, and the mean energetic 
distance from the earth's center greater. When the muzzle- 
velocity had attained to some 26,000 feet per second the elliptic 
orbit would have become widened into a circle, clearing the 
earth's surface and its mountain-tops, and returning the pro- 
jectile to the point of its start once each eighty-five minutes. 
If the cannon had been wheeled out of the way during the 
progress of the first circuit of the earth, and if the atmosphere 
were absent, the projectile would continue thus as a satellite of 
the earth, in circular orbit, indefinitely.* 

Suppose now that the muzzle-velocity of the projectile might 
be still further increased. Its orbit will now have become elliptic 
again. But now the point of start is located, not at apastron, but 
at periastron of the orbit. The projectile, at apastron, is ''over" 
the antipodes ; and at each increase of muzzle-velocity its height 
above the antipodes increases. The eccentricity of orbit now 
increases again. The major axis of the elliptical orbit elongates, 
and the period of time elapsing between each two returns of th^ 
projectile to the platform increases, according to Kepler's Third 
Law, from the original eighty-five minutes by the three-halves 
power of the major axis. 

Finally, when the muzzle-velocity should have reached and 
surpassed some 37,000 feet per second, the orbit would have 
become, first parabolic, and then hyperbolic; and the projectile 
would depart from the earth forever. Any higher muzzle- 



*This velocity is derived from Equation 17. Taking the values for 
g as 32.18, for the radius of the orbit as 3,960 miles and the mass of the 
earth as something less than 42X10^^, it appears from Equations 7 and 17 
that the velocities of equilibrium are as follows : — 

Ft. per sec. 

Circular motion around the earth, just above its surface, 26,000 

Vertical motion, directly "up" or away from the earth, suf- 
ficient to carry the projectile indefinitely into space — or contrari- 
wise, the velocity of striking the earth after free fall from space 
directly toward the earth's center, 37,000 

The last figure gives the maximum velocity of relative motion which 
can exist between a mass-portion of the density and dimensions of our 
earth and a mass-portion considerably smaller, as the exclusive property 
of the pair, and which can continue indefinitely, without dissipation in 
either collision or dissociation. Should hisrher velocities than this be 
observed they must be the property not of the mass-oair between which 
they are observed by the eye, but of some much larger mass-system. 
Only such a larger mass-system could either generate or arrest such 
excessive velocities. 



64 ENERGY 

velocities than this would, of course, only accentuate the prompt- 
ness of this dissociation. 

For any given mass-pair — and virtually, for any given major 
mass, when the individual masses are widely dissimilar — this 
"critical" velocity at periastron, above which dissociation takes 
place, is a definite thing, for any given periastron distance So. 
For, as was noted in the preceding paper, the expression for the 

spacial intensity of energy of a pair is c (^ c" ) j ^^^ if, in this, 

S becomes very great indeed, the intensity takes the definite 

c 
value -^. Therefore, if two bodies fall together from any dis- 

tance of separation whatever, however great, into a minimum 
separation of So, this is the utmost intensity of motion which 
they can develop. In the case of the earth this velocity is about 
37,000 feet per second. Conversely, two bodies situated at the 
minimum distance of So require only this intensity of motion to 
dissociate them permanently. 

This degree of intensity, then, is the utmost which this par- 
ticular pair is capable of embodying, unless conditions permit 
them to approach nearer than the distance Sq. Ordinarily, So is 
limited by the finite dimensions of the bodies, when solid ; but 
theoretically there is no limit to the diminution of So, and as it 
diminishes the intensity of energy increases very rapidly. 

But at present it is not so important to discuss the possible 
variations of So as it is to note that, if any pair be observed at 
periastron with an intensity of motion greater than the ''critical" 

. c . . . 

one, corresponding to-^, implymg a hyperbolic orbit and prompt 

dissociation, it signifies that the pair visualizes what it cannot, 

c 

of itself, embody, viz : a fund of intensity greater than ^— * 

The surplus of energy over this quantity cannot possibly be the 
property of the pair itself. It is a manifestation by the pair of 
an energy-fund held by it in conjunction with some third ex- 
ternal mass-portion, which third portion is itself not necessarily 
visible directly as a member of the system. But nothing but 
some such larger third mass-portion is capable of destroying the 



CRITICAL ENERGETIC CONDITIONS 65 

surplus velocity, and none but it could have been capable of 
producing it. 

This surplus, or third-mass, energy may be called, for the 
present, the ''external" energy of the pair, in order to keep our 
argument in terms of a simple mass-pair as a base. This ex- 
ternal energy the pair has borrowed and displayed as its own, 
so to speak, although it has no rightful ownership of it and must 
soon repay the loan. 

The relation of such external obligations to its own internal 
assets of energy is visible directly in the value of the eccentricity 
of orbit e. For it is only when e is greater than unity that 
hyperbolic motion and dissociation can occur. The excess of e 
above unity, therefore, is a measure of the external or borrowed 
energy which is displayed by the pair. 

In order to understand this situation completely, return must 
be made for a moment to the question of radial and tangential 
energies. 

A value of e greater than unity implies an angle between 
mean energetic motion and the radius vector connecting the two 
bodies (see Fig. 4) smaller than 45°. In that case the radial 
component of motion would be greater than the tangential. 
HerCj again, one must be careful with his mathematical ex- 
pressions for energy. If it be assumed, too readily, that kinetic 
energies are absolutely proportional to the square of the velocity, 
then the ratio of radial to tangential kinetic energies, at the mean 
energetic point, must be e"^ ; for ^ = cotan a, the ratio of radial 
to tangential velocity at this point. 

But two objections exist to this reasoning. In the first place, 
no exact expression for the tangential energy exists at all. In 
the second place, if there were one it must be negatively, rather 
than directly, proportional to the velocity squared ; for energy 
must be abstracted in order to increase the tangential velocity. 

The 'way out of this puzzle is to note, from Equation 22, 

Radial Energy = 2 c M.M^ -^ = 2 e M,M, -^^^^^ (22) 

that, for any stated mean energetic conditions, such as distance 
D and tangential velocity U sin a, the radial energy is directly 
proportional to the eccentricity of orbit e. And since the mass- 
pairing factor M1M2 is always alike on both sides of this equa- 



66 ENERGY 

tion, bearing no influence upon the result, it would have been 
more explicit if this statement had been made in terms of 
intensities of energy, rather than of energy itself. Thus it may 
be considered proven that when the radial intensity — which has 
already been characterized as the medium of communication, or 
manifestation, between the pair and the external world of mass — 
assumes a proportion to the internal, or stay-at-home, intensity 
greater than equahty, then dissociation and a cessation of the 
pair's visible existence must ensue promptly. It is with a mole- 
cule much as it is with a man : when he goes too energetically into 
foreign politics, to the overshadowing of his private business, his 
home breaks up. 

The Lower Critical Intensity. It has already been sug- 
gested, however, that something else than dissociation may 
happen to interrupt the continuity of the pair's energetic existence 
in smooth, unbroken orbit, with energy manifestly conserved. 
Collision may occur, at or just before periastron. 

For this to take place, the distance of separation at periastron 
must be less than the sum of the radii of the two bodies. The 
situation must therefore be investigated through the energetic 
equation in terms of periastron conditions, or Equation 2^. 

E,=2cM,M,^-l-- = 2M,M,^^-^. (23) 

The last two terms of this equation readily reduce to 

c V^ 1 

which is the fundamental equation for the critical intensity, in 
which the first two terms define the limits of spacial concentra- 
tion, on the one hand, or of embodiment of internal motion, on 
the other, which respectively constitute the criteria of energy- 
transformation. 

If, in Equation 27, So be considered equal to the sum of the 
radii of the two bodies, the expression gives the limiting condi- 
tion by which collision at periastron may be avoided. 

The Critical Limits of Intensity. Upon these considera- 
tions may be founded the statement that the determining factor 
in the permanence of energy is its intensity. Within certain 
limits the intensity of a given pair may vary ; with effect upon the 
external appearance of the energy, it is true, but without pre- 



CRITICAL ENERGETIC CONDITIONS 67 

cipitating a transformation of the energy into some other appar- 
ently independent form. But as soon as these limits are sur- 
passed, the energy alters its outward aspect so radically as to 
make it difficult to discern the true continuity of its existence. 
We are constrained to say that the energy has therein become 
"transformed," so that we give it another name than before. 
And as the understanding of the more obscure forms of energy 
is inextricably connected with energy-transformation, which gives 
them birth and death, it is clear that the entire science of ener- 
getics turns upon these critical limits of intensity as a car does 
upon two king-pins. 

These limits, and their effects, may be stated briefly as 
follows : 

1. If the kinetic, or outward radial, intensity exceed the 
limit implied by the condition e=i, the pair will dissociate. 

2. If the spacial, or inward radial, intensity exceed the 

limit implied by the condition -—= wherein the R's are 

the radii of the two masses, the pair will collide. (For the 
intensity increases, it will be remembered, as So grows smaller.) 

Contrasting these two sorts of critical condition, in terms of 
the contrasted aspects of the two extremes of the simple orbit, 
the first is perceptible as a critical limit to the spacial expansion 
of the system, by too great velocity radially outward — in which 
phenomenon force plays little part. The second is perceptible as 
a critical limit to the forceful compression of the system, by too 
great a velocity radially inward — in which phenomenon space 
plays little part. It will develop later that the first form of 
critical condition is of interest in connection with the too great 
expansion of vapors and gases. The second is of interest in 
connection with the too forceful compression of solids. For 
either procedure will develop energy-transformation in bodies of 
matter. 

After dissociation occurs, the further history of the pair is 
about as easily written as is that of the snakes in Ireland. Vir- 
tually speaking, there is no further history. The pair has ceased 
to exist, as a pair, though its two members continue an inde- 
structible existence. 

Yet in this facile statement, which is necessary in order to 
take the road step by step, lies in reality a great error against 



68 ENERGY 

which the student should be warned. In truth, however dis- 
tantly (and apparently permanently) a mass-pair may be sepa- 
rated, by chance foreign influence, yet it is never completely 
sundered, and never can be. Though the distance of separation 
may become that which now lies between any object at hand 
and those scattered throughout the farther heavens, at distances 
beyond human comprehension, yet the mutual bond of attraction 
still exists, in the case of each pair, to a measurable degree. 
And it exists eternally. Though the number of centuries which 
must elapse, before chance again frees the pair for a fall into 
mutual propinquity and perceptible energetic reaction, may far 
surpass human understanding, and strain even our arithmetic to 
compass its expression, yet when the time does come the latent 
mutual affection will be just as potent for warmth of greeting 
as if it occurred to-day. 

As to the further results of collision, on the other hand, that 
topic lies much nearer to the purpose of these papers than does 
dissociation. Indeed, its bearing is so vital that it will be post- 
poned for a special chapter of its own. In the meantime, atten- 
tion may be turned to those processes which may lead the energy 
of any mass-pair to exceed the critical limits of kinetic or spacial 
intensity, with the results just defined. 

Collision and consolidation of the pair into unity may result 
from the abstraction of energy in either of two ways, viz : 

1. If the energy be extracted in radial form (that is, between 
periastron and apastron, by a medium capable of absorbing only 
the radial component of motion), the eccentricity e decreases. 
Because the tangential component has remained unaffected, the 
mean energetic distance D will remain unaltered. The motion 
becomes more nearly circular, and thus lapses toward a peaceful 
consolidation ; for it was explained in an earlier paper that true 
circularity of motion constituted an apparent unity. 

2. If the energy be abstracted in tangential form (that is, 
at apastron, by a medium capable of absorbing only tangential 
energy), the eccentricity e is increased while the mean energetic 
distance D is decreased. From both reasons the periastron dis- 
tance So decreases. The motion becomes more nearly rectilinear, 
and may impinge upon the solid confines of the bodies. Thus 
would ensue a violent consolidation of the pair. 

Thus, in the illustration of the cannon-ball, velocities at 



CRITICAL ENERGETIC CONDITIONS 69 

periastron varying only between about 26,000 and 37,000 feet 
per second would permit permanency of orbit. Anything higher 
would lead to dissociation ; anything lower to collision. If, when 
velocities were between these critical conditions, the radial com- 
ponent only of motion should be abstracted, the orbit would 
reduce toward a circle of large radius about the earth, at a rate 
of 26,000 feet or less per second. The speed mentioned would 
then be the one of equilibrium, and the pair would exist per- 
manently, as a ''unit/' in this condition, until again disturbed 
from without. 

But this process of abstracting energy radially could never 
be carried to the point where the orbit became truly circular, 
with zero eccentricity. For energy might be abstracted radially 
only to the extent that a radial component of motion were 
present. As the orbit approached circularity radial energy could 
be abstracted only with greater and greater difficulty, or under 
more and more unusual conditions. The process could not imag- 
inably proceed until all eccentricity were gone. 

If, on the other hand, the energy were abstracted tangentially 
at apastron, or at any rate during the outer half of the elliptic 
orbit, the latter would become a more narrow ellipse, like that 
of any cannon-ball, and collision with the earth would ensue 
quite as in any such a case. 

These actions can perhaps be better understood from Fig. 5, 
which displays the several possible orbits of a body M^ relatively 
to a larger mate. Around the mate is shown a dotted circle ZZ, 
the radius of which is the sum of the radii of the (supposedly 
spherical) bodies. Any orbit which touches this circle will of 
course end in collision. Thus, AA and FF are both hyperbolic 
orbits ; but one of them will end in dissociation, the other in 
collision. 

Should radial energy be abstracted from AA by radial means, 
between P and A, the orbit would be altered to some hyperbola 
or ellipse of less eccentricity than AA and passing through the 
point M^ ; that is, the mean energetic distance would be con- 
served. The limiting case for such a process, when all the 
radial energy had been withdrawn, would be the circle, which 
is the only one of these orbits of lesser eccentricity which is 
shown. 

The withdrawal of tangential energy from AA would lead to 



70 



ENERGY 



a hyperbola of increased eccentricity, passing through M^ and 
cutting the circle ZZ. 

At periastron P the radial energy of AA (as was pointed out 
in an earlier paper) is all kinetic in form; and that kinetic 
energy is directed tangentially. Yet all of it above the velocity 
for circular equilibrium at this radius is true radial energy; for 




its radial or centritugal force is unbalanced, and the motion- 
energy will quickly become radial in direction as well as name. 
Therefore, if kinetic energy be abstracted at P, it amounts to a 
reduction of radial energy. Radial energy will have been 
reduced, though not by radial action. The eccentricity will 
decrease and the orbit finally become an ellipse, such as BB. 
Further abstractions of kinetic energy at P will alter the orbit 
to a circle, such as CC. or an ellipse of reversed eccentricity, 
such as EE, which ends in collision. 

The abstraction of kinetic energy at the apastron Q of BB, 



CRITICAL ENERGETIC CONDITIONS 71 

on the other hand, increases the eccentricity, but decreases the 
periastron distance, as in the orbit GG, and leads to colHsion. 
For at Q the kinetic energy is all tangential energy, the radial 
energy being space-energy. 

Now in the illustration of the cannon-ball the energy was all 
supposed to be supplied at periastron, as at P for the orbits CC, 
BB or AA. But in the use for which these arguments are 
designed, as mechanical similes for molecular action, all contribu- 
tions or abstractions of energy will be by other systems which are 
external to the one in question. The point of contact between 
the two systems will be at or near apastron, such as Q for BB or 
GG, or P for EE. The only way in which tangential energy 
can be exchanged is tangentially and kinetically ; and even then 
it can be exchanged only through the medium of conversion into 
or from radial energy. 

Radial energy, on the other hand, cannot only be exchanged 
directly with external systems, but it can be exchanged in two 
distinct and different ways. These ways are (i) spacially, by 
radial action between periastron and apastron, and (2) kinet- 
ically, by tangential action at periastron or apastron. Referring 
to the illustration of the cannon-ball, the first would be instanced 
by the interaction of the projectile, during its outward or inward 
flight, with some mass which was itself moving outwardly or 
inwardly, relatively to the earth. The second would be in- 
stanced by the original impulse of the projectile, by the gun- 
powder, or by some similar, but negative, influence experienced 
from some third mass at the extremity of its flight away from 
the earth. 

This question of the possible ways by which energy may be 
gained or lost by a mass-pair should be clearly understood. 
They may be stated briefly as follows, although their significance 
will not come out until the discussion reaches the question of the 
mechanical theory of heat. Thus, energy may be gained or 
rejected by a mass-pair in three different ways, each of which 
will have a different effect upon the form of orbit and upon the 
chances of its alteration past one of the critical points of 
intensity. 

I. Energy received tangentially at apastron (as in the illus- 
trative cannon-ball before its trajectory had cleared the earth) 
increases the mean energetic distance D and decreases the eccen^ 



72 ENERGY 

tricity e. Energy imparted tangentially at apastron, of course, 
reverses this rule. 

2. Energy received tangentially at periastron (as in the 
illustrative cannon-ball after its trajectory had cleared the earth) 
increases both the mean energetic distance and the eccentricity. 
Energy lost in similar fashion reverses this rule. 

3. Energy received radially, between one extreme energetic 
condition and the other, increases the eccentricity, but does not 
affect the mean energetic distance. Energy expended radially 
decreases the eccentricity without aft'ecting the mean energetic 
distance. 

In the illustration of the cannon-ball, by which the natural 
action of a free projectile was sought to be made clear, there 
was used, it is true, an original impulse, viz : the energy of the 
gunpowder, which is foreign to the simple interaction of two 
mass-portions by gravitation and inertia. Yet the illustration 
may not, for this reason, be outlawed ; for in the interaction of a 
projectile with another mass-system consisting of more than one 
mass-portion there may arise a form of impulse which closely 
resembles that of gunpowder and cannon. 

If we revert to the simple energetic system displayed in Fig. 
2 and imagine M2 to be, instead of a doughnut-like solid, a ring 
of separate solid mass-portions^ revolving about the center C in a 
swarm, in circular orbits, then the situation becomes more com- 
plex, and capable of activities beyond those already described. 
Let it further be imagined that Mj, instead of following the 
straight line ACB back and forth, follows some conic-section 
orbit which carries it periodically through the point C. 

Now the intensity of action at this point was found to be 

proportional to -5-, wherein So is the minimum distance of 

approach between M^ and Mg. But when M, is an annular body 
such as shown in Fig. 2, the minimum value for So is the radius 
of the annulus. Therefore, if the annulus should contract dur- 
ing any of Mj's circuits of its orbit, the intensity of action at its 
next approach to M2 would be accentuated. The energy released 
from M2 by the mutual approach of its members would be 
transferred to Mj, imparting to it a greater velocity; and this 
increase, occurring at periastron, must be radial in its character. 



CRITICAL ENERGETIC CONDITIONS 73 

It will go to increase both the mean energetic distance and the 
eccentricity of orbit between M^ and Mg. 

If the argument be transferred to Fig. 3 new possibilities 
appear. If M^ thereof should consist of two or more portions, 
held apart by their relative motion in elliptic orbit, the degree of 
propinquity at periastron, or the intensity of energy of the 
MjMg-system, would depend upon whether the two or more 
orbits all reached periastron simultaneously or not. Should Mg 
be in its condition of maximum separation of parts when M^ 
approached, the intensity of action would be slight. But if all 
orbits coincided in phase, reaching greatest propinquity simulta- 
neously, the intensity would be very much greater. The impulse 
received by M^ in any such a way would be quite the parallel 
of the impulse received by the illustrative, cannon-ball from the 
gunpowder, at periastron. 

Thus, from period to period of the M^Mg-system the 
intensity of its radial energy might vary widely. If M-^ were the 
messenger carrying a manifest of the system's energy to foreign 
parts, which could not directly perceive the more massive, tan- 
gential and torpid energy of M2, the system would be observed 
thereby as embodying widely varying intensities of energ}^ 
While at one time the intensity might be so limited as to confine 
Mj^ to an elliptic orbit, at another it might project it with an 
hyperbolic orbit. Or, conversely, the projectile M^, having entered 
the system from foreign parts on an hyperbolic orbit, might 
be entrapped and remain in elliptic motion. Indeed, its own 
arrival might be the cause of such dissociation between the parts 
of Mg, at the expense of M/s energy, that the latter no longer 
possessed sufficient energy to get away again. 

Of the portions of the orbit further from periastron, those 
beyond the mean energetic condition are most illustrative ; though 
ail that is said also applies to points inside the mean. 

In its outer portions the orbit may be perturbed by the 
approach of some third mass-portion arriving as a messenger 
from more distant field-s. This third projectile will possess a 
motion aimed more or less directly at the mass-center of the 
MiM2-system. To the extent of its component thus directed it 
will exchange radial energy with the system. To the extent of 
its component normal to this direction it will exchange tan- 



74 ENERGY 

gential energy. The effects of these exchanges upon the form 
of the original orbit have already been noted. 

It is not possible to trace here all the varied possibilities 
of such interchanges of energy, even when occurring between 
systems of only three or four mass-portions. What is desired 
is merely to impart an elementary concept of the distinctions 
between radial and tangential energies, and the ways in which 
these may be affected by other systems without infraction of 
the Conservation of Energy. It is now plain that such outside 
influences might alter widely the intensity and eccentricity of 
any energy-system, leading it toward or over either of the two 
critical limits of intensity as defined above. It will be under- 
taken later to show that such outside influences may also affect 
just as widely the extensity of a mass-system, though the process 
is not quite so simple and obvious. 

This completes the bare statement of what mechanical energy 
is, and by what methods only it may be imagined as augmented 
or diminished. Whatever hypotheses as to other forms of 
energy than mechanical may be made, if the latter are to be 
regarded at all as mechanical energy in disguise their activities 
must be brought into line with the preceding analysis. 

For the present it suffices to point out that there has already 
resulted from this analysis a useful classification of mechanical 
energy-forms, which may be given a habitation and a set of 
names in the following table, as "permanent," *'subpermanent" 
and *'superpermanent" types of energy, respectively. 

1. The permanent forms of mechanical energy are those 
embodied in elliptic orbits of sufficient dimensions to clear the 
solid confines of the two bodies. The mathematical conditions 
defining this class are that the eccentricity e shall be less than 
unity and the periastron distance So shall be greater than s, when 
z is the sum of the radii of the two solid masses. 

2. The subpermanent forms of mechanical energy are those 
embodied in orbits, either elliptic or hyperbolic, which pass 
within the solid confines of the bodies and end in collision. The 
mathematical conditions defining this class are that e may have 
any value whatever, but So must be less than z. 

3. The super permanent forms of mechanical energy are 
those embodied in hyperbolic orbits only, when of sufficient 



CRITICAL ENERGETIC CONDITIONS 75 

periastron distance to clear the solid confines of the bodies. 
They end in dissociation, instead of collision. The mathematical 
conditions defining this class are that e shall be greater than unity 
and So greater than z. 

It will develop later that, whereas all the so-called "me- 
chanical" energies of the engineer must belong solely to the 
second of these three classes of energy, the molecular or atomic 
or electronic energies, which we call heat, chemical energy, 
electricity, etc. — if they can be regarded as modes of mechanical 
motion at all — must belong to the first and third types. As 
to electrical energy there is more doubt, because lack of 
permanence is its chief characteristic ; yet in electrical matters 
all questions of time must be referred to such exceedingly minute 
units that it may appear, upon examination, that electricity, like 
light, is in reality one of the most permanent of all forms of 
energy. 

In the distinctions portrayed in Fig. 5, therefore, we are on 
the edge of understanding that most wonderful and significant 
of all natural phenomena, the transformation of energy. For, 
aside from the contrasts between the kinetic and potential, and 
the radial and tangential — sorts which form a part of every type 
of energy — no alterations in form of action have developed, thus 
far in the analysis, which are so subversive of external appear- 
ance as are these changes from permanent to sub- or super- 
permanent conditions. 

The dividing lines between the permanent, as the central form 
of energy, and the sub- and superpermanent forms on its either 
hand, are called the critical energetic conditions — the lower and 
upper critical intensities of energy, respectively. These critical 
limits have already been briefly defined in this chapter. The 
part they play in energetic phenomena cannot be discussed until 
other forms of energy than the mechanical are discusssed. 

The upper critical intensity has long been recognized and 
taught — although chiefly of interest in astrophysics — as the 
"critical velocity." For each mass-portion, it has been taught, 
there exists a certain velocity for smaller mass-portions, above 
which the latter would dissociate from the former. If this idea 
be made more accurate by defining the critical function as a 

■y2 

certain value of , instead of merely velocity, and 

M1+M2 



76 ENERGY 

broadened by recognizing it as an intensity of motion-energy, it 
can remain for present purposes unchanged. 

The lower critical intensity, on the other hand, has not been 
similarly recognized and taught, so far as the ordinary text- 
books give evidence. It is directly proportional to intensity of 

space-energy, or degree of propinquity, or solidarity, ., accord- 
So 

ing to the several ways in which it might be named. For every 

mass-pair of specified density of members, therefore, there is a 

certain degree of intensity of concentration (which is itself a 

form of energy) which cannot be exceeded without entailing 

collision and energy-transformation. 

In all of this discussion it must have become long since 
obvious that, in the natural, mechanical interaction of mass- 
portions, the straight line plays a very subordinate part, if it 
appears at all. As a matter of fact, it does not appear at all. 
It has already been clearly shown how, mathematically speaking, 
the straight line constitutes one unattainable limit of eccentricity 
of orbit, on one side, while the circle constitutes another equally 
unattainable limit of eccentricity (namely, zero eccentricity) on 
the other. Centrally between the two lies the parabola, with 
unit-eccentricity, constituting the geometrical fundament of nat- 
ural energetics. It would be the next step of development of 
this question of the absorption and rejection of energy by mass- 
systems to show that this mathematical aspect of the situation is 
also the natural one — except that in mathematics limits are 
attainable, whereas in nature they are not. 

For it is a fact, in the natural aspect of the question, that the 
more radial energy a system possesses, the more readily it will 
reject or impart energy to other systems, and the less readily it 
will receive it in radial form. Conversely, the more tangential 
energy a system embodies the more readily it will receive and 
absorb radial energy, and the less readily it will impart it. It 
follows, therefore, that the smaller the eccentricity becomes, the 
greater difficulty is there in reducing it further, and the greater 
are the chances that the eccentricity will increase, in any ener- 
getic mix-up, rather than still further decrease. Conversely, the 
greater the eccentricity becomes, the greater is the likelihood of 



CRITICAL ENERGETIC CONDITIONS 77 

its being decreased, in any energetic encounter, rather than being 
still further increased. 

The result of this view of the case is to place the medium 
value for the eccentricity — unity — in the most conspicuous posi- 
tion, as the value forming a center of stable equilibrium, on 
either side of which the variations in eccentricity swing, as does 
a pendulum about its vertical position. Just as it was found (see 
Chapters II and III) that both velocity and spacial separation, in 
energetic systems, swing in stable equilibrium on either side of 
mean energetic values for both variables — on either side of a 
mean energetic distance of separation and a mean energetic 
velocity, neither of which could ever attain to either zero or 
infinity — so now it appears that the eccentricity of natural orbit 
also varies on either side of its mean energetic value. 

This mean energetic eccentricity of orbit is unity, defining 
the parabola. This is the mean or average energetic condition 
of every mass-pair in the universe. A mass-pair in this mean 
condition of eccentricity would be just upon the line between 
confining its energies at home, in stable permanence, and sending 
them abroad. From this mean the eccentricity may be reduced 
into elliptic motion by the abstraction of energy from the system. 
But such abstraction of energy becomes more and more difficult 
as it proceeds, and no imaginable natural conditions may ever 
be defined which would succeed in reducing the eccentricity to 
zero, in circular motion. 

Similarly, the absorption of energy by any mass-pair having 
the mean energetic eccentricity of orbit, or following a parabolic 
orbit, may increase that eccentricity into hyperbolic motion 
indefinitely. But the difficulty of further absorption of energy 
increases as it proceeds, and no imaginable natural conditions 
may be defined which would succeed in expanding the eccentricity 
to infinity, or developing straight-line motion. 

It is therefore obvious that the teaching of mechanics to 
mature students by basing everything upon straight-line motion is 
unnatural to the last degree. The fundament of all true mechanics 
is parabolic motion. That stands to all other forms of orbit as 
our sun stands to all possible vagaries of its innumerable family 
of satellites — as a natural base and center of equilibrium which, 
albeit itself unsupported and undefined in space, may yet never be 
disregarded as the natural starting point for all discussion. 



CHAPTER VI. 

The General Nature of Mechanical Energy. 

The definition of mechanical energy is now complete, so far 
as a definite, though skeleton-like, structure is concerned. But 
this skeleton needs clothing with some flesh and form, before it 
may Le useful for a display of the nature of heat. 

The outline of the skeleton, to summarize for convenience, 
may be stated as follows : 

1. Energy has been identified as always consisting of the 
arithmetical product of two variables. One of these variables 
has been named the intensity, and the other the extensity, of 
energy. 

2. Intensity has been shown to be a function of either the 
.y/'flc^-relationship or the 7woffon-relationship between two or 
more mass-portions. The intensity of space-relationship is pro- 
portional to the "propinquity," or the reciprocal of the distance 
of separation. The intensity of motion-relationship is propor- 
tional to velocity-squared-divided-by-aggregate-mass-involved. 

3. Extensity has been shown to be the measure of the 
amount of mass-pairing involved. It is proportional, other things 
being equal, to the square of the total mass involved. For any 
given total mass it increases, but not proportionally, with the 
degree of subdivision of that mass, into mass-pairs capable of 
embodying the relationships defined above. 

4. It was shown that energy-quantities may vary by varia- 
tions in either intensity or extent. In the applied mechanics of 
engineering it is only the intensity-factor which varies appre- 
ciably; that is to say, we vary space or motion, while the mass- 
factor remains proportional to the mass involved. But in thermal 
energy or other intricate forms, when viewed mechanically, the 
extensity-factor must be expected to vary as often and widely 
as- the intensity ; that is to say, the energy, its intensity being 
fixed, is no longer necessarily proportional to the mass involved. 

5. The variation of any mass-system in intensity of energy 
may take place smoothly, in stable equilibrium, within a certain 

78 



GENERAL NATURE OF ENERGY 79 

range. This range is defined at either end by the two critical 

intensities. Trespass over either critical intensity causes the 

equilibrium to become unstable. Trespass over the lower, or 

.... . . c 

spacial, critical limit of propmquity, or — , leads to collision. 

z 

Trespass over the upper, or kinetic, critical limit of intensity, or 
, leads to dissociation. Either collision or dissociation 

constitutes a transformation of energy. 

6. Mechanical energy existing in stable equiHbrium, between 
the critical limits of intensity, may be called "permanent" in 
form. That embodying intensity greater than the spacial limit 
of propinquity has been called *'subpermanent" in type. That 
embodying motion above the upper critical limit of kinetic in- 
tensity has been called "sup'erpermanent." Of the latter two, 
the first type can exist only throughout a portion of a single 
revolution, and is perceptible to the human senses only when the 
members of the mass-pair are large enough, and the period of 
revolution long enough, for their separate observation. Such is 
the case in celestial mechanical energies, and in the applied 
mechanics of machines. 

7. It is next to be pointed out — and this is one of the most 
important steps in the understanding of energy — that the ener- 
getic conditions of matter, whether spacial or kinetic, whether 
referring to intensity or extensity, never spring from, nor are 
measurable from, an absolute zero of any one of the factors 
involved. Instead, the factors in any energetic condition vary 
on either side of a central, or mean energetic, value; and this 
value itself hangs self-supported in space, so to speak, with no 
means known for referring it to any absolute base. It is to an 
explanation of these statements that Figs. 6 and 7, and the next 
few paragraphs of discussion, are to be devoted. 

The earlier papers of this series defined the intensity of 

energy as proportional to -^ ^, when potential, and of 

bo O 

V2 — V 2 . . 

^ — I — ~- when kinetic. The law of the conservation of energy 

Mj -j- Mg 

links these two forms, so that either may be studied as a repre- 
sentative of both. 

Energy may consist either of little space and much motion 



80 ENERGY 

(or force), or of much space and little motion (or force). But 
these paired quantities appear, not as a sum, but as a product. 
If they appeared as a sum, either of them could be reduced to 
zero at times (the quantity of energy remaining constant, accord- 
ing to the conservation of energy) by a sufficient growth of 
the other. But being bound together as a product, neither factor 
may be reduced to zero by any finite growth, however great, of 
the other. And since "infinities" apply only to portions of the 
universe so large as to exceed human understanding and meas- 
urement, and therefore have no place in any exact natural sci- 
ence, zeros are likewise excluded from participation in energetic 
phenomena. 

Thus, as space disappears, in nature, energy of motion 

appears in proportion to , and force appears in proportion to 

space 

). Therefore no finite accumulation of energy or force, 

space y 

however great, can ever make the space zero, or compress matter 
into nothingness. This agrees with our most ordinary concepts 
of matter; for two of the prime attributes of matter, defined as 
elementary in the earliest study of nature as an exact science, 
were its indestructibility and its occupancy of space. 

On the other hand, as space appears force disappears, and 
energy is absorbed. Yet no imaginable degree of finite space 
can ever reduce the force to zero, or quite annul the absorption 
of energy with further increase of space. This concept dates 
from Newton's discovery of the law of gravitation, and lies at 
the heart of our modern concept of the universe as a unit — its 
every part bound inseparably to its every other by an unbreak- 
able, albeit a very elastic, bond. This concept is most familiar 
to engineers in connection with gases and vapors, which expand 
indefinitely, losing pressure as they go, yet with no possibility of 
the pressure ever reaching zero. 

As for velocities, they must follow the same general law, 
although in accordance with a different mathematical function. 
Whereas force is proportional to the square, and energy to the 
first power, of the propinquity, or the reciprocal of space, 
velocity is proportional to the square root of that same function. 
Although the rates of variation would therefore differ in these 
three cases, yet the general form of the relationship remains the 



GENERAL NATURE OF ENERGY 



81 



same. No velocity can be so great as to reduce the space to 
zero, and no space so great as to reduce the balancing velocity 
to zero. 

This general form of relationship between force, energy or 
motion, on the one hand, and space on the other, is shown in 
Fig. 6. The curve would not be the same in the three cases, but 
it would have the same general form; and because of the diffi- 
culty of making one scale show all three functions to advantage, 
one only is shown to represent them all. The function is seen 
to be a curve asymptotic to the two rectangular axes. Each 
factor may vary as widely as it pleases, and may thereby vary 
the other. But neither can ever force the other to zero, by 
increasing ever so widely. 
S„ 



COMPRESSION 



FORCE AND MOTION 



1=^ 



FIG. 6. 



Moreover, as either may seek to force the other to increase, 
by itself decreasing toward zero, it will find itself working 
against an increasing mechanical disadvantage as it proceeds. 
The further it goes the greater is the proportion of resultant to 
creative action. By the principle of virtual velocities, further 
progress must become more and more difficult with each advance. 
The tendency is always to return toward a central or medium 
value for each of the factors. When space becomes deficient 
and the force excessive, force tends to control the situation ; as 
a compressed spring or gas tends to burst its bonds. When 



82 ENERGY 

space, becomes excessive and force deficient, space tends to rule 
the game; as when an elevated weight tends to fall, or a dis- 
tended gas to be condensed by external pressure. The natural 
equilibrium in which these factors vibrate on either side of their 
central, or mean energetic, values, in either direction, is thor- 
oughly stable. 

This same general form of energetic relationship and sta- 
bility of equilibrium applies also to the other energetic variables, 
as well as to space, force and motion. Thus, in Fig. 7 can be 
seen the way in which the extensity-factor, or quantity-factor, 
of energy varies in terms of the degree of subdivision of the 
aggregate mass embodying energy of any stated intensity. In 
the Fourth Paper was developed the equation for this relation- 
ship, in Equation 26, w^hich is repeated here 

X = JM' (1-i-) (26) 

for convenience. In it X is the extent of mass-pairing, or ex- 
tensity of energy, which is embodied in the mass M by its sub- 
division into any number n of equal parts. 

The variation of X with n, for any given mass, is seen in 
Fig. 7. When n = i, or the mass is a homogeneous, solid unit, 
embodying one arbitrary unit of mass, X=o and the function 
appears at A, Fig. 7. When n=2, X=iM^ and the curve 
passes to D, upon a scale determined by the size of the mass- 
system or the arbitrary mass-unit in question. But as n 
increases still further, X exhibits an increasing slowness in fol- 
lowing proportionality to it. A doubling of n to 4 increases X 
by only one-half. A quadrupling of w to 8 increases X by only 
three-quarters ; and no finite extension of the value of n, however 
great, can quite succeed in doubling the value of X from the 
point D. 

When the arbitrary unit of mass which forms the measure 
of each "equal" portion becomes greater than one-half the total 
mass — that is, when the aggregate mass is divided into only two 
mieqnal portions, the larger one of them constituting the unit of 
mass and the fractional remainder — n may have values (always 
positive) which are less than two. Such would be the case in 
engineering mechanics, where from the total mass of the earth 
only a small fraction is split off, made into a hammer or a 
cannon-ball or a locomotive, and its energy relatively to the 



GENERAL NATURE OF ENERGY 



S3 



remainder (which we still call ''the earth") utilized for human 
purposes. In such case X would become a small fraction 
of M-, and the curve of Fig. 7 would pass from D toward G. 
When the total mass present amounts to just one unit of mass, 
X becomes equal to zero. When the total mass present is less 
than one unit of mass, n becomes less than unity and X becomes 
negative. 

As the left-hand limb of the curve approaches the condition 
of a straight line parallel with the axis, the degree of mass- 
pairing, or the extensity of energy, approaches proportionality 
with the mass of the smaller fragment. Absolute proportionality 
is what is assumed in the equations employed in engineering. 




EXTENT OF MASS-PAIRING 



FIG. 7. 

But it is plain from Fig. 7 that this assumption could become 
true only in the impossible case when n became zero, when the 
fragment split off from the earth became zero and the extensity 
of energy became minus infinity — for only then would the hmb 
AC of the curve BAG have become a straight line. 

It therefore becomes plain that the extensity of a mass- 
system, or its capacity for embodying intensity of energy, varies, 
with the fineness of its subdivision into separate portions, quite 
as does space with motion. The relationship swings on either 
side of a central, or mean energetic, condition, or arbitrary zero, 
such as D, Fig. 7. No absolute zero is attainable in either 
direction. Even if the axes which may be said to measure 
absolute zeros, at the foot and right-hand, respectively, be 



84 ENERGY 

regarded as basis of convenience which it would be well to retain, 
the fact constantly to be kept in mind is that the energetic 
condition never passes to either of them, and never can. This 
general characteristic holds true, whether expressed in terms of 
intensity or extensity of energy, whether of space, motion, force, 
degree of massive solidarity on the one hand, or of fineness of 
comminution of mass on the other. Any of these factors may 
pass to either very great or very small values, but none of them 
may ever attain to either zero or infinity. 

Indeed, it will appear, as the argument proceeds, that every 
energetic relationship which can be stated exactly follows this 
same general law. To those engaged in power-engineering the 
most familiar illustration of this statement is the hyperbolic 
relation between the pressure and volume of any gas. Pressure 
and volume always appear as a product, each being inversely 
proportional to the other, or to some power of the other. The 
general equation is PV^=a constant. They never appear as a 
sum, one decreasing as the other increases. No degree of pressure, 
however great, can ever reduce the volume of any gas to zero; 
nor can any degree of expansion, however great, ever reduce the 
pressure to zero. There is no place in the universe where the 
pressure or density or volume of elastic matter is imaginably zero. 

In every case, all energetic functions are founded upon a 
central, or mean energetic, condition, which hangs unsupported 
in space, so to speak, as the sun hangs in the heavens. No abso- 
lute base or support for it is imaginable or necessary. It is on 
either side of this central point of reference, and not up and 
down from any absolute zero, that all energetic factors vary. 
These statements, which have been made in reference to 
mechanical energy only, will be found to apply universally. 

Energetic Equilibrium. In all natural phenomena the one 
most important guiding principle, after conservation, is that of 
universal stability of equilibrium. The determination of what 
shall be the next in that most intricate series of occurrences to 
which we give the general name, the progress of events, always 
depends upon stability of equilibrium. The natural universe is 
always, except locally and temporarily, in stable equilibrium. 
And if its equilibrium temporarily and locally has become un- 
stable, the movement is always toward the recovery of stability. 



GENERAL NATURE OF ENERGY 85 

Whatever may occur in the nature of a departure from the 
general medial trend of affairs always brings with it, as its 
immediate consequence, a tendency to departure in the counter- 
vailing or balancing direction. Although this tendency may not 
prevail immediately, it must ultimately. 

This law has its foundation in these elementary mechanical 
systems now under discussion. They were likened, in the open- 
ing pages of the second paper, to the familiar pendulum, which 
is seen to swing always in stable equilibrium. Turning from 
that to the less familiar, but only true, energetic element, the 
two-part free mass-pair, the same truth appears. The element 
swings in stable equilibrium between two extremes, one of un- 
usual space and the other of unusual force or motion. In this 
swing, departure in either direction begets increasingly a ten- 
dency to return. The attainment of unusual space kills the 
motion which begot it, and increasingly invites motion of return. 
Unusual lack of space begets both force and velocity, and tends 
increasingly to a reversal of motion and a recreation of space. 

The same law applies to the mass-pairing factor of energy. 
It varies on either side of a central value, of a mean average 
size of solid or undivided mass-portion, in stable equilibrium. 
The unusual consolidation of any numiber of the mass-portions 
of a system begets unusual disgregative velocity in the 
remainder. This of itself constitutes a dispersion of matter. But 
in addition, the unusual velocity of this remainder, returning in 
due time, tends to beget a renewed separation of the originally 
consolidated group. As much as this can be seen in pure 
mechanics, with collision and heat-formation excluded from the 
discussion; but so soon as these phenomena are admitted, as 
will be done in the next chapter, the field of this form of stability 
of equilibrium will reveal its extension into other forms of 
energy, in a most beautiful way. 

The same law applies to the eccentricity of orbit. Unusual 
eccentricity tends to impart energy radially from the system to 
outside bodies; and the loss of this energy tends to a reduction 
of the eccentricity. Unusual lack of eccentricity, on the other 
hand, invites the absorption of energy contributed radially from 
other systems ; and the effect of such absorption is necessarily to 
increase the eccentricity. 

Obviously, too, this phenomenon cannot proceed to any rigid 



86 ENERGY 

or abrupt limits. Eccentricity of orbit can never reduce itself 
quite to zero, by the radiation of energy; because the ability to 
do so depends upon the presence of the eccentricity itself. Cir- 
cularity of orbit cannot absorb radial energy to the point where 
the eccentricity is infinite, because the ability to absorb is lost 
as the eccentricity increases. 

The fundamental law of this equilibrium, as evinced between 
eccentricity, mass and dimension of orbit, is based upon the 
conditions found to prevail in our solar system; wherein the 
few score bodies, the motions of which we can study, have had 
time, since the dawn of astronomy at least, to settle into stable 
equilibrium. The existing state of affairs is defined in the 
equation derived independently by La Place and Lagrange, viz : 

MM 

- ' ' i, .6^ VS + S = a constant* (28) 

This is not an equation of energy-interchange, but one of fact, 
showing the effect of centuries of energy-interchange. It is 
incidental to and illustrative of our argument, rather than basic 
for it. But it is of more than incidental significance that this 
equation reveals the same general relationship between the 
factors of energy as those given previously. The three factors 
of mass, space and eccentricity are linked together, not as a 
constant sum or a constant ratio, but as a constant product. 
Any one of the three remaining constant for the time, either of 
the other two can vary to an unlimited degree; hut only as the 
reciprocal of the third. Neither can be brought to zero by any 
expansion, however great, of the other. Neither can approach 
zero without encountering increasing resistance, in the unusual 
expansion of the other which must accompany it. 

In all of these respects, the elementary free mass-pair does 
not constitute for engineering students a forcible illustration, for 
here on the earth's surface we have no free mass-pairs big 
enough to be seen. All that we have which are free are of 
molecular dimensions, and our knowledge concerning them is 
chiefly inference. But as consideration turns to the forms of 
energy other than mechanical, it will appear that all energetic 



*The writer Is uncertain whether the first factor in this equation 
should be as printed, or simply M, Consistency with all the other true 
equations of mechanics would give it the form here printed. But, like 
all these other basic equations, it is to be found under the cognisance of 
high authorities in terms of simple M. 



GENERAL NATURE OF ENERGY 87 

action follows, in a most striking way, these same general char- 
acteristics. All of them swing constantly on either side of a 
mean energetic condition, against resistances which increase as 
the departure from the mean condition increases, between limits 
of zero and infinity neither of which can ever imaginably be 
reached. It is of vital importance to the mechanical theories of 
heat and these other energy-forms, therefore, that these same 
basic characteristics be noted as attributes of the most ele- 
mentary, mechanically energetic mass-pair. 

Energy-transformation. So far as the mathematical forms 
of the curves connecting the several factors of energy are con- 
cerned, these swings of energetic condition on either side of the 
central mean might extend indefinitely, along the asymptotes to 
either axis. But when mathematics is replaced by observation 
of natural fact, it appears that each curve fails of continuity, if 
pushed too far along its asymptote. Some factor hitherto irrele- 
vant enters and controls the situation. The energetic equilibrium, 
stable up to this point, becomes abruptly unstable. Smooth inter- 
action at a distance between the two mass-portions comes to an 
end. Either dissociation enters, to put an end to the identity of 
the mass-pair as a perceptible pair, or collision enters to put an 
end to the conservation of the original form of energy. 

This, then, is energy-transformation, the break in the con- 
tinuity of the curves of stable equilibrium and of visible con- 
servation of energy which reveal the critical limits of intensity 
of energy — the critical limits to the concentration of energy in, or 
of abstraction of energy from, a mass-system of the particular 
degrees of mass and of mass-pairing in question. 

What ensues then is more difficult to explain than what has 
preceded. We know now, from considerations broader than any 
yet permitted to enter the argument, that trespass beyond these 
critical limits of intensity abrogates neither the Conservation of 
Energy nor the universal Stability of Equilibrium. It is only in 
terms of the particular form of energy in question — in this case 
mechanical energy — that the continuity of conservation and sta- 
bility is broken. The line is then crossed which arbitrarily 
defines this energy-form from the others ; and across this line, with 
the energy, we must step, if we are to follow clearly the con- 
tinuity of universal natural action. When we have crossed we 
shall see that what we have crossed was indeed an arbitrary line, 



88 ENERGY 

like a state boundary-line, erected in the human imagination to 
serve the convenience of human limitations ; but having no 
other real existence. We shall see that, so far as light now 
penetrates, all energies are one in their fundamental components. 
The sole trouble is that the light does not penetrate far. In 
other energy-forms than mechanical we cannot see these com- 
ponent parts. We know these more obscure energies only by 
their blanket results. Yet if we are ever to get any more clear 
and concrete idea of their anatomy, it seems inevitable that it 
should be in terms of mechanical energy. Certainly the engineer 
and the engineering-student, if not all others, can proceed more 
clearly from concepts based upon mass, space, force and motion 
than they can by relying solely upon abstract empiricisms, stated 
mathematically. The writer will attempt no argument that heat 
is or is not a "mode of motion." He believes that it is. The 
evidence that it is, albeit vague and inconclusive to some minds, 
is too great in volume for denial. We should lose too great a 
portion of our scientific perceptions if we should deny the me- 
chanical nature of heat, and do it consistently. But that is not 
the point. The point is that if heat be truly a "mode of motion" 
and chemical energy truly a ''mode of arrangement" of mass in 
energetic action, our concepts thereof must be guided by the 
fundamental principles of mechanical energy which have been 
displayed above. We are now, for the first time in the argument, 
mechanically equipped for an accurate pursuit of the questions : 
What mode of motion? What mode of arrangement? among 
the many imaginable ones, may or must they be ? 



CHAPTER VII. 

What is Heat? 

In asking ourselves what is heat the most surprising thing is 
to think that not every one, that possibly no one, knows what 
heat is. Heat is one of the commonest things in the world. It 
is as common as matter ; for we know of no matter without heat. 
It is as common as space; for while space cannot embody heat, 
yet no space is known which does not contain either matter or 
else that radiant energy (commonly called "light") which, trav- 
eling always at the inconceivable rate of 186,000 miles per 
second, turns into heat as soon as it meets solid matter. 

Yet we have no definition of heat. Heat seems to be many 
different things, according to how it is encountered. It has been 
called ''the waste-heap of the universe." Indeed, it seems to be 
easier to say what heat is not, than what it is. Heat is not 
matter. That point was settled a century ago. It is a ''form of 
energy." But all that that means is that it is capable of per- 
forming work. But as it is capable of many other things besides 
performing work, and as many other things besides heat are 
capable of performing work, this is not a very satisfactory 
definition. 

Heat is like an ant-colony. It lives in a hidden nest. We can 
see it go into and come out of its nest — matter — by several 
doors. But as to just whither those doors may lead, and what 
may be the form of structure which connects the several doors, 
is as yet pure surmise. 

Fortunately, there are quite a number of doors to the thermal 
ant-hill ; and as heat appears at each of these it bears a different 
guise. So that, aided rather than hindered by the very diversity 
of the problem, we are able to guess fairly near to the sort of 
interior arrangement which alone could fit all of the doors. 

Heat appears in and disappears from matter by the following 
processes : 

89 



90 ENERGY 

Methods of Heat-gain. Methods of Heat-loss. 

(i) Conduction from hotter bodies. Conduction to colder bodies. 

(2) Absorption of radiation. Radiation to colder bodies. 

(3) Impact and friction. ■. 

(4) Compression. Expansion. 

(5) Combustion, Dissociation. 

(6) Electrical resistance. Electrical generation (by thermo- 

pile). 

There are other thermal processes than these, but they occur 
upon too small a scale to be of present interest. 

Of all of the above processes the two most familiar sources 
of heat are radiation (sun-heat) and combustion. But both of 
these processes are complex and obscure, when viewed from the 
stand-point of the present articles, which are to concern them- 
selves with a mechanical explanation of heat. Sun-heat is a 
transformation of radiant energy, and combustion a transforma- 
tion of chemical energy, into heat; and both radiant energy and 
chemical energy are just as much in need of an explanation as is 
heat itself. 

Mechanical work is the only form of energy of which we 
now have any definite and clear concept. It is by the doors 
opening between that form and heat that the latter must be 
approached. These doors are the processes numbered three and 
four; and of these Number Three comes first, both numerically 
and naturally. 

But the question of impact and friction can be broached for 
discussion only in terms of elasticity and its opposite. 

Elasticity and Inelasticity. When two solid bodies come 
into contact the collision is always partially elastic and partially 
inelastic. That is to say, a part of the kinetic energy inherent 
in the bodies before collision is returned, in the form of motion 
in the reverse direction, and a part is not. In so far as the 
energy is returned kinetically, the bodies are said to be elastic. 
In so far as it is not, they are said to be inelastic. While some 
bodies are almost perfectly elastic, and others almost wholly 
inelastic, none are known which are completely either. 

In so far as bodies are elastic, their collision can have no 
effect upon the mechanical principles laid down in the preceding 
papers. Two bodies engaged in a mutual orbit which brought 
them into collision would, if perfectly elastic, rebound with a 
velocity as great as that before collision. Only the direction of 
motion would be altered. The original conic-section orbit would 



WHAT IS HEAT? 91 

be continued unaltered, except that its new axis would be 
inclined with its old one. The mean energetic and critical condi- 
tions would remain at the same intensities; but the collision 
which constituted the lower critical intensity would not lead to a 
transformation of energy, as when inelasticity is present. 

Elasticity, however, while incapable of throwing any light 
upon the principles of motion, throws considerable light upon 
the energetic nature of mass — at least, in the form of the so- 
called "solid" bodies. For elastic collision means the temporary 
storage of the energy of collision in the deformation of the 
bodies, against their disposition to retain their solid form ; which 
stored energy is given out again in the rebound, as the original 
forms are regained. 

But, if our ideas are to remain true to the elements of 
mechanical action as stated by Kepler, Newton and La Place, as 
collocated in the preceding papers, this temporary storage of 
energy within each body cannot be attributed to pure mass. It 
is only in changed relationships between mass-portions that 
energy can be stored. Each of the elastic colliding solids must 
therefore be regarded, not as a unified or truly solid portion 
of mass, but as a more or less complex system of mass-portions, 
between which the energy may be stored. The way in which 
this may be done is not the point of immediate interest. The 
significant fact is that no body which exhibits any elasticity 
whatever, may be regarded as a solid unit. N'o truly single or 
homogeneous body, whether it be of the magnitude of a moon 
or a molecule or an electron, can possess elasticity, any more 
than it can possess energy. 

Elasticity can be an attribute only of the subdivision of mass. 
Wherever the mind may be disposed to chase that most elusive 
concept, "the ultimately indivisible portion" of truly solid or 
homogeneous mass, the one quality which must be assigned to it 
is that of perfect inelasticity. The attempted concept of an 
ultimately indivisible, yet perfectly elastic, "atom" is as incon- 
sistent with all accurate scientific experience and principle as is 
the concept of perpetual motion. Both concepts have arisen 
from the desire for a royal road of unnatural ease to the solution 
of natural problems. 

As for inelasticity, that brings the discussion home to the 
question of what is heat; for inelasticity is merely a short name 



92 ENERGY 

for the degree to which kinetic energy is converted into heat by 
impact, when bodies colHde. 

In order to get the problem into simple form, let it be sup- 
posed that the colliding bodies are perfectly inelastic; for the 
addition afterward of that modicum of elasticity which is always 
present in fact will not affect our conclusions as to the portion 
which is inelastic. Now the word ''heat" being merely a sub- 
terfuge, or cloak for ignorance, with which we cover up the 
fact that the energy disappears and we do not know what form 
it takes, let heat be excluded from the discussion. We may 
refuse to use the word until we have an exact idea to attach to it. 

What form, then, may the energy of the colliding bodies take, 
if both elastic rebound and the formation of heat are excluded, 
and the conservation of energy is still to hold true? Only one, 
by any possibility, viz : the rupture of the bodies and the separa- 
tion and scattering of their fragments. This is the only truly 
mechanical process, aside from elastic rebound, which will absorb 
energy inelastically. 

But, if the bodies collide as free bodies, uninfluenced by the 
propinquity of other and greater masses, such as the earth, the 
fragments will not stay scattered. They will fall together again. 
Because of the increase in mass-pairing by the splitting up of 
the original bodies, and also by the loss of energy in their 
fracture, the average intensity of the fragment-pairs must be 
less than that of the original pair; and since the bodies were led 
to collide, the fragments will do likewise, to a partial degree at 
least. 

But these secondary collisions between the fragments occur 
under the same conditions as did the primary collision. There 
is to be no elasticity, and no shrouding of the energy under the 
mysterious term "heat." Therefore the secondary collisions can 
result only as did the primary, viz : in a further subdivision or 
comminution of the fragments. And these secondary fragments 
m.ust again collide, in tertiary collisions, etc., etc. 

To this process there can be but one inevitable end. Col- 
lision, fracture and disgregation must take place again and again, 
although with diminishing violence, until a condition of per- 
manently stable equilibrium is reached by the fragments 
becoming so small that they no longer collide. Instead, they will 
have come, one after another, as each became small enough, to 



WHAT IS HEAT?" 93 

adopt elliptic or hyperbolic orbits of revolution about one 
another, without collision, in permanently stable equilibrium and 
with energy perfectly conserved. Their subpermanent energy 
will have become of the "perrnanent" type. Their inelasticity 
will have become elastic, not by some miraculous metamorphosis 
from ordinary matter into molecular matter, but merely by 
foregoing contact at all, procuring reversal of motion by force- 
action *'at a distance," instead of by collision supposed, in 
violence of all natural experience, to be perfectly elastic. 

But this permanency of energetic condition is just what is 
called ''heat," viz : a permanent mode of motion-energy and 
space-energy between the particles of a body, resultant from 
inelastic coUision. The prime characteristic of heat is its per- 
manence. All other forms of energy apparently tend to turn 
into heat, pretty completely, at all times, while the heat tends 
to remain heat. That is why heat has been called *'the waste- 
heap of the universe," and the prediction has been freely 
indulged in that ultimately all other energy in the universe must 
become heat. 

The writer would explicitly avoid giving countenance to so 
extreme a belief as this. Yet undoubtedly the prime charac- 
teristic of heat is its permanence and stability of equilibrium, 
when considered as a result of and in contrast with the abrupt 
instability of the mechanical energy of solids and liquids moving 
in contact. Temporarily, at least, the energy has reached a 
permanence of form which must extend over a period covering 
many millions of the vibrations of the very tiny mass-pairs 
embodying the heat — far too long to permit the hypothesis of 
collision occurring. For there is no collision known to science 
which is not somewhat inelastic and dissipative of its energy. 

The attempt to define heat has therefore accomplished its 
first stage. Heat is a mode of motion and of separation among 
a swarm of tiny fragments of the mass of the hot body, none 
of which possess subpermanent orbits. 

As to superpermanency of orbit, discussion of the possibility 
of that being a part of thermal interaction must be deferred. 
Other questions as to the form of these intermolecular orbits 
and interactions must also be deferred, until the many diverse 
p'^culiarities of heat may have been gotten more clearly in view. 
To this end good use can be made of the thermal diagram. 



94 ENERGY 

And when all the data are thus displayed it may appear that 
several of the other sources of heat, which are much more 
obscure in form and nature than impact and friction, have 
characteristics so like to these that this entrance into the thermal 
ant-hill through only one of the many doors may not seem so 
one-sided and inconclusive a plan after all. 



CHAPTER VIII. 

The Thermal Diagram. 

Let heat-producing impact and friction be imagined as occur- 
ring against a specific weight, such as one pound, of some 
famihar soHd, such as ice. Let it be imagined that the ice were 
originally at the very lowest temperature of which scientific 
investigation has had experience, w^here the ice would be a very 
cold, hard, brittle solid. The result of the impact and friction 
would be to raise the temperature of this solid; and if it were 
continued sufficiently, it would ultimately melt the ice and carry 
the resultant water through all the thermal experiences of which 
the substance H2O is capable. 

It will be of convenience to represent this process graphically. 
And since the previous analysis has shown energy always to 
consist of the product of two independent variables, it will be 
most natural to regard thermal energy also, from the start, as 
made up of the arithmetical product of two variables. Indeed, 
if heat is to be considered as a "mode of motion," or one form 
of mechanical energy, at all, it must be considered as the arith- 
metical product of two variable factors; for this constitution 
was everywhere found to be a prime characteristic of mechanical 
energy. 

When any quantity thus consists of the product of two 
variables, it is most conveniently represented as an area, upon a 
field of rectangular coordinates. The two independent factors 
then become the two coordinates, respectively. But if heat is to 
be depicted thus, the identity of the two coordinate factors stands, 
at the start, as a matter of guess-work. An easy first guess for 
one of them is temperature; for temperature has been, from the 
beginning of thermal science, recognized as a prime factor in 
heat. And yet it has also long been known that temperature is 
not heat. 

If this first guess has been a true one, in accord with the 
natural facts, then the second coordinate (regarding which noth- 
ing can be known at the start) will prove to be identical with 

95 



96 ENERGY 

some natural prime factor in thermal phenomena also. But if 
the first guess should prove to have been wrong, then the whole 
graphical situation will fall into chaos, in a reductio ad absurdum. 

But if a fair knowledge of thermodynamics on the part of 
the reader be assumed, it will be plain that the selection of tem- 
perature as one of the prime factors of heat is no wild, irre- 
sponsible guess. It is now more than eighty years since Carnot 
proved conclusively that temperature was the one fundamental 
feature of heat in the guidance of work-performance; and that, 
too, by pure empiricism, without attempting any definition of 
either heat or temperature. It is now more than a quarter of a 
century since Lord Kelvin defined the only true temperature- 
scale in terms of work, rather than of heat; and since Maxwell 
proved, in the mathematical theory of the so-called ''perfect 
gas," that temperature was the translational kinetic energy of 
the flying particles of thermal matter, and therefore a real 
physical quantity. 

But the writer wishes especially to avoid humbugging both 
himself and the reader by starting from premises which are laid 
down too rigidly, as if they were absolute law. For in this whole 
field of discussion we possess no such rigid premises — unless the 
laws of Kepler and Newton, which are now unquestionable, be 
such. The ideas as to temperature, heat, entropy, etc., must fit 
the facts ; that is all. While the present discussion has started 
rigidly enough from the exact mechanics of Kepler, Newton and 
La Place, because their work has stood the tests of centuries, yet 
the entire hypothesis that heat is "a mode of motion" at all, it 
must be remembered, yet hangs unsettled in mid-air. If it can 
be made to appear that what exact data we possess as to heat fit 
what exact data we possess as to mechanics, heat may be accepted 
as a mode of motion. But until that is settled our premises must 
remain assumptions and guess-work, and should be defined 
clearly as such. 

With this to start with, the thermal diagram may be put 
under construction, as at B, at the lower left-hand corner of 
Fig. 8. The vertical axis of this diagram is to measure "abso- 
lute" temperature, along the axis OT. This locates two hori- 
zontal axes: one at XX for the absolute zero of temperature, 
and the other at ZZ for the Fahrenheit zero. 

Areas, then, are to measure heat, and heat as supplied by 



THE THERMAL DIAGRAM 97 

Impact and friction. But the other coordinate of the diagram 
is for the present unknown. Therefore it must be defined in 
terms of the two quantities which are known. Plainly, it must 
be the result of dividing area (or heat) by height (or tem- 
perature). 

But in doing this it must be remembered, as was stated in 
the opening pages of the First Paper, that energy is a name for a 
change of condition only, and not for something absolute. There- 
fore, since conditions change constantly with increments of 
energy, our definition must be confined to exceedingly small 
increments of energy at a time; or, in short, must be stated as a 
differential. 

Let the horizontal coordinate be given the symbol N. Then 
our stated premises are defined mathematically by Equation 29, 

dN=^ = K-^ (29) 

wherein dQ signifies the quantity of impactive energy absorbed, 
T the absolute temperature of the body at the time of impact, 
and K the specific heat of the body.* 

Starting therefore at B, Fig. 8, the curve which represents 
the thermal experiences of the ice under impact must be some 
such an one as BC, simultaneously rising in temperature and 
departing to the right, with positive values for dN. The equation 
for this curve can be had by integrating Equation 29, which is 
easily done if K the specific heat be a constant. The result 
then is 

N-No = Kloge-^ (30) 

wherein the zero-subscripts refer to any original, and the 
unmarked symbols to any final, condition. Should the specific heat 
be not a constant, the form of the curve would be slightly altered 

*The writer has no desire to impose upon the reader the suggestion, 
from the above language, that the thermal diagram thus developed, which 
will prove to be identical with the well known entropy-temperature dia- 
gram, is originated in these pages. But the ultimate significance of this 
diagram has been so doubtful and obscure that the writer wishes to avoid 
its adoption in the premises. The argument has been arranged to be self- 
contained. No dogmatic assertions have been made in the premises, 
except the mechanics of Kepler and Newton and the principles of 
the conservation of mass and energy. If the conclusions appear to be 
too faulty, or too extensive to be easily accepted, the blame will not be 
lost amid a haze of ill-defined or too rigidly assumed premises. 



98 ENERGY 

and the integration not so easy; but the general conclusions 
would be unaltered. 

During this warming of the ice by impact there occurs, appar- 
ently, no change in its physical state. The ice remains a 
crystalline solid. Only its temperature alters, together with the 
yet undefined horizontal factor of heat. From this fact the 
process and curve BC, with Equation 30, have been given the 
name isomorphic — the syllable *'morph" indicating form and the 
prefix ''iso" the fact that it remains unchanged. The term is 
used in contrast with metamorphic, which is applied to those 
processes, such as fusion, vaporization, etc., where the physical 
state of the body does undergo a material change in form. 

From Equation 30 it is clear that if the original temperature 
To from which the ice was warmed be imagined as occurring at 
lower and lower points, the point B must be regarded as moving 
indefinitely to the left, as it approaches the "absolute zero" Hmit 
of temperature. The curve BC must be asymptotic to this axis, 
as it trends to the left. No finite value of N — No can be great 
enough to make To^o, when the value of T is finite. That is 
to say, the absolute zero of temperature is absolutely unattainable 
in nature. It is as unreal as is any absolute zero of either space, 
motion, force or mechanical energy. 

As the isomorphic BC proceeds to the right, however, it finds 
an abrupt termination at the point C. When a temperature of 
^2° F. is reached the ice begins to melt. Further supplies of 
impactive or frictional energy continue to be absorbed, and it is 
to be inferred that they take the form of heat. This fact can be 
easily proven by other experiments, such as mixing the ice with 
red-hot iron, when the iron will be cooled and the ice melted. 
But the latter would not be warmed. It would absorb the heat 
given up by the iron as "latent" heat, in a change of physical 
state at constant temperature. 

Such a process would be shown in Fig. 8 by the straight hori- 
zontal line CA. It is horizontal because the temperature does 
not change during the heat-absorption. It is straight and dotted 
because it does not represent a continuous process, but a gap, 
across which jumps molecule after molecule of ice, as it acquires 
heat enough, in unstable equilibrium. 

It is known that the amount of heat absorbed in this process 
is large (142 B.t.u. per pound). It is known that the melting 



THE THERMAL DIAGRAM 99 

consists of a breaking up of the ice-crystals into formless liquid. 
The volume of resultant water is less than that of the ice ; 
nevertheless, there is reason to believe that the space actually 
occupied by the substance itself increases during fusion. The 
ice-crystals may be likened to a cord of fire-wood; when con- 
verted into saw-dust less space will be occupied than by the pile 
of logs, with their many interstices ; yet if these interstices be 
deducted from the original volume, the space occupied by the 
wood has increased. A few large interstices have been 
exchanged for many small ones — the former being distinguishable 
from the substance, but the latter not. 

Such a process as this melting of the ice is called meta- 
morphic, implying change of form, in contrast with the isomor- 
phic processes such as BC The metamorphic processes are all 
virtually isothermal. The heat which they absorb is called latent, 
because imperceptible by the thermometer, in contrast with the 
sensible heat absorbed in the isomorphic processes. 

Equation 29 easily becomes 

dQ=TdN (31) 

If the relation between T and dN be known, as by knowing the 
specific heat, this equation can be integrated into an expression 
for Q — Qo, the quantity of heat involved in passing from any 
one condition to any other. The quantity dQ is shown in Fig. 8, 
as a very narrow vertical rectangle, having a height T and a 
width dN. The integration of this rectangle will develop the 
area beneath the curve bounding the upper ends and limited at 
right and left by the ordinates of original and final conditions. 
Thus, the heat required to warm the ice from the condition B to 
the condition C will be given by the area BCc. That required 
to melt the ice will be given by the area CAOc. 

The melted ice at the point A constitutes the arbitrary zero 
of our steam-tables. If the process of adding energy by impact 
and friction be continued, the water will renew its rise in tem- 
perature, this time along the isomorphic ADD'W. At the same 
time its vapor-tension, or the pressure which is exerted at all 
temperatures by the vapor struggling to free itself from the 
water, increases. 

If this vapor-tension should happen to equal the pressure 
exerted upon the water by the surrounding objects by the time 



100 



ENERGY 



the temperature D is reached, the molecular equilibrium again 
becomes unstable. The water cannot absorb more energy in the 
shape of internal motion without its internal centrifugal pressure 
exceeding the external centripetal pressure; wherefore it must 
burst. So burst it does, molecule after molecule, into steam, as 
fast as each molecule becomes hot enough; just as pop-corns 
do in a corn-popper. 

The popping of each molecule alters its condition suddenly 
and completely from that shown at D to that shown at E. There 
is no stability of equilibrium between D and E, and no molecule 
may stop part way after having once started across. If the 
thermal condition of a quantity of water and steam is ever 
shown at any intermediate point, such as P or Y, it means not 
that the entire pound of molecules is in such intermediate con- 
dition, but that one-half or less are in the condition D and the 
other half or more are in the condition E. 




dH c Q a 



FIG. 8a. 



THE THERMAL DIAGRAM 



101 



The latent heat of vaporization is shown 
by the area beneath DE, or dDEe. It is much 
greater than the latent heat of fusion, as 
shown by comparing this area with CAOc. 
In this case there is no doubt as to the 
increase in volume. The volume of each 
molecule after "popping" is several hundred 
times that before, the ratio depending upon 
the pressure under which vaporization occurs. 

Indeed, this increase in volume is so great 
that it expands into visibility an energy- 
quantity which has hitherto been negligible. 
This is the external work. The energy 
supplied to each molecule along DE con- 
sists not only of that required to burst 




THE MINIMUM LIMIT OF TEMPERATURE 

FIG. 8b. 



102 ENERGY 

the molecule against its own internal bonds of unity, called the 
disgregation-workj^ but the external forces must be pushed back 
also, throughout a considerable increase in volume. Yet in this 
case the external work amounts only to about one-ninth to one- 
fifteenth of the disgregation-work, so powerful are the congre- 
gative molecular forces. 

If the pressure upon the heated water had been higher than 
that permitting vaporization at D, that process would have been 
delayed until some higher temperature had been reached, as at 
D'. Therefore, there may be as many different vaporization- 
levels as there are different pressures. The ends of these various 
metamorphic lines, such as DE, D'E', etc., form a curve SS which 
is called the saturation-curve. But the student is especially 
warned against thinking of the saturation-curve as representing 
a process, as do most of the other curves in the thermal diagram. 
There is no process known to the boiler or engine rooms which 
will convert saturated steam of one pressure and temperature 
into saturated steam of another pressure and temperature. It 
takes a combination of at least two processes to do this, and an 
impossibly delicate balance of the two, at that — unless the steam 
be in contact with water with which it may interchange heat 
promptly, in which case the heat is added not merely to the 
steam, but to the water also.f 

If the addition of energy by friction and impact to the pound 
of HoO at E be continued still further, the steam will rise in 
temperature again along the isomorphic EF of superheated 
steam. The volume increases almost proportionally with the 
temperature. The disgregation-work has now become the minor 
portion of the energy absorbed, the external work being in 
ascendancy. 



*"Disgregation" implies the scattering of a flock, the opposite of "con^ 
gregation," the gathering of a flock. 

fFor this reason the use of the term "specific heat of saturated steam,'* 
meaning the difference in the total heats of two points on the saturation- 
curve separated vertically by one degree of temperature, cannot be too 
strongly condemned, as loose and misleading. The term "specific heat" 
is used generally and properly to signify the quantity of heat which, added 
to a substance, will raise its temperature by one degree, the physical 
state remaining constant. The use of the term "snecific heat of saturated 
steam" therefore leads the student to infer, most naturally, that tf you 
add to saturated steam containing the total heat Hi the heat H2 — Hi, 
wherein H2 is the total heat of saturated steam one degree hotter, there 
would result saturated steam of the total heat H2 and temperature Ti-|-I. 
Yet nothing could be further from the truth. 



THE THERMAL DIAGRAM 103 

At some higher temperature, such as F, there arises a new 
condition of unstable internal equilibrium within the molecule. 
If energy continues to be added, molecule after molecule of the 
steam bursts again — but this time not into a new ''physical" 
state of H2O, but into a new "chemical" state of dissociated 
hydrogen and oxygen. The energy now absorbed in potential 
form, along the metamorph EG, is some seven times as great as 
that absorbed in vaporization ; just as that absorbed in vaporiza- 
tion was some seven times that involved in fusion. Moreover, 
it is not called latent thermal energy, but chemical energy. For 
this practise there are excellent reasons; but it is one of the 
offices of Fig. 8 to show plainly how much more closely allied 
are heat and chemical energy than is commonly supposed. It 
is also to remind the reader that chemical energy is distinctly a 
latent form of eftergy. 

The increase in volume which is involved in this dissociation 
amounts to only fifty per cent. ; and the external work involved 
amounts to less than two per cent, of the whole. Therefore it 
may be said that virtually all of the energy absorbed goes into 
latent, chemical or disgregative form. 

Should the process of adding energy by impact — which has 
now become most difficult, because of the almost perfect elas- 
ticity of the substance — be persisted in, the temperature again 
rises along an isomorph, GH; and to this isomorph there is no 
further interruption, by instability of internal equilibrium, so 
far as the writer is aware. Moreover, because of the high 
temperature already attained, the horizontal departure dN in- 
volved in the absorption of a thermal unit dQ has become so . 
small, and the corresponding rise in temperature has become so 
great (because the specific heat is much less than for liquids or 
solids), that the isomorph GH rapidly becomes approximately 
asymptotic to some limiting vertical axis, not yet exactly located. 

Should the oxy-hydrogen mixture which is illustrated as 
active in the isomorph GH be cooled again, by the conduction 
of its heat to colder bodies, it may be said (though the statement 
is unsupported by what has preceded) that it will return along 
the curve HGFEDACB which it came up. But if the two gases 
be separated, and then cooled, their thermal condition will be 
shown by the curve HGRh. 

If, at any comparatively low temperature such as h, the 



104 ENERGY 

gases be mixed again and then heated, when they reach their 
temperature of ignition, as at R, they will again combine, 
releasing chemical energy which must be absorbed thermally. In 
order that the combination may be complete, the thermal energy 
must be absorbed from the substance, by some outside body, 
bringing the mixture again into the condition F. The sum of 
these processes may be illustrated by the curve RF, although an 
intermediate passage of some portions of the substance into the 
condition H is involved. 

In the curve hRH is visible a smooth transit of a substance 
from one thermal condition which is asymptotic to the axis of 
absolute zero of temperature, to another thermal condition which 
is asymptotic to an axis at right angles thereto. In the curve 
BCADEFGH is visible a somewhat similar path of thermal 
transit, broken by localities of unstable equilibrium, it is true, 
but terminating in each direction in a smooth curve of stable 
equilibrium, which is asymptotic to horizontal and vertical axes, 
respectively. The field visible between these curves is crossed, 
from curve to curve, back and forth, by the familiar processes 
of melting, freezing, vaporizing, condensing, combustion and 
dissociation. The curves themselves represent the even more 
familiar processes of warming and cooling. 

If it be remembered that every substance known, except a 
few gases rare even in the chemical laboratory, occurs in all of 
the three physical states: solid, liquid and gaseous, and is subject 
to chemical dissociation and combination, in endothermic and 
exothermic processes; and if it be remembered that all these 
processes, for all these substances, might be represented upon 
Fig. 8, with no departure from what is already there, except in 
the number ^nd confusion of lines and in the choice of con- 
venient scales ; and if it be further remembered that all the 
processes for conversions between heat and work, as well as 
those between heat and chemical energy, may be displayed upon 
this diagram (though as yet there has been no reason to refer 
to some of them) — it becomes evident that Fig. 8 places before 
the eye in comprehensive form a pretty complete picture of the 
data which are concerned in and essential to a complete under- 
standing of the relationships existing between heat, work and 
chemJcal energy. If there be any truth in the hypothesis that 
heat is indeed a mode of motion, and chemical energy a mode 



THE THERMAL DIAGRAM 105 

of arrangement, it should certainly come out from a thorough 
inspection of Fig. 8, in terms of the analysis of mechanical 
energy already accomplished. 

This task must be deferred for later papers. For the present 
it is desired merely to call attention to the general fact that the 
curves of energetic equilibrium displayed in Fig. 8 have the same 
general form as those displayed in the Sixth Paper, covering 
all known cases in purely mechanical energy. That is to say, 
they vary, on either side of a central, or mean energetic, con- 
dition, which central condition is not definitely located from any 
rigid or absolute base, indefinitely toward and along two axes 
(at right angles with each other) to which they become asymp- 
totic and which they can never reach. Unlimited departure from 
either axis is natural and imaginable; but unlimited approach 
toward either one is not. While it is true that the upper limb 
of the heat-curve of Fig. 8 is not a true asymptote, as it is there 
drawn, yet its similarity to one is obvious. Whether or not it be 
a true asymptote will be discussed more in detail later. 



CHAPTER IX. 

Mechanical Concepts of Thermal Phenomena. 

a. pressure and volume. 

If it be assumed that the preceding papers have supplied 
complete data for the understanding of work, heat and chemical 
energy, in so far as the last named is related to the other two, 
the first task is to construct therefrom mechanical concepts of 
the four fundamental attributes of matter which are active in 
these fields, viz : 

(a) Pressure, 

(b) Volume, 

(c) Temperature, and 

(d) Entropy. 

These are all thermal attributes. Pressure and volume are ther- 
modynamic in character, bridging the gap between heat and work. 
Temperature and entropy are almost purely thermal attributes, 
but also are active in thermochemical phenomena. 

Besides the above attributes, it is also necessary to explain in 
mechanical terms, two fundamental processes, viz: — 

(e) Heat-development and transfer, and 

(f) Thermodynamic work-performance or work-absorption. 

Of all of these phenomena, according to the writer's view, 
the one easiest to comprehend is volume, and the hardest one 
is pressure. Both are inextricably mixed up with heat-action ; 
and yet the occurrence of mechanical action where pressure 
and volume are involved, yet where no heat-changes are per- 
ceptible, is common. But in all cases, if the facts be closely 
examined, both pressure and volume will be found to be thermal 
phenomena. For no substance can be imagined as reduced to a 
zero of either pressure or volume without first being carried to 
that unattainable thermal condition, the absolute zero of tempera- 
ture. 

106 



MECHANICAL CONCEPTS 107 

Volume. If the idea that heat is a mode of motion is to be 
adhered to consistently, the volume exhibited by any body must 
be the effect of the separation between its component mass- 
particles. Since the internal condition of the body is permanent 
and stable, these mass-particles must be in a free condition of 
natural equilibrium. If so, their relative separation, in the face 
of their mutual attraction toward each other, as well as in the 
face of external pressure, must be maintained by motion of 
revolution about each other. This is the only possible me- 
chanical explanation of the occupancy of space by elastic matter. 

In order to explain volume alone, this internal motion might 
be regarded as purely circular in form; and this is the simplest 
idea with which to start. In that case, the centrifugal and 
centripetal forces would be exactly balanced, and there would 
be no active exertion of expansive pressure outwardly. There 
would be, however, a passive, and at times a stalwart, resistance 
to compressive pressures acting radially inwardly. 

Since the motion of the particles (under the above assump- 
tion) is purely tangential, its velocity must increase as the 
volume of the body becomes smaller. But during any such a 
change, where circularity of motion is conserved, the energy 
cannot be conserved. Energy must be abstracted in order that 
the volume of the body should become smaller and the tan- 
gential velocities greater. 

The only special provision to be made, in imagining the 
volume of a body as made up of a vast number of tiny mass- 
pairs, revolving as described in the earlier papers upon me- 
chanical energy, is that these many orbits must lie in all sorts 
of planes, interacting at oblique angles, whereas the elementary 
orbits studied were all specified as confined to a single plane. 
But this provision introduces no new principles of action. 

Pressure. The above hypothesis furnishes no explanation 
of pressure active outwardly. For this can be explained only 
upon the assumption of eccentricity of orbit between the par- 
ticles, involving as it must a lack of balance between centrifugal 
and centripetal forces at both apastron and periastron. Indeed, 
this lack of balance is true of every point of an eccentric orbit 
except the mean energetic point; and even there, although the 
forces are balanced, there exists an unbalanced fund of radial 



108 ENERGY 

motion, acting outwardly on one side of the orbit and inwardly 
on the other. 

Now all known conditions of matter exhibit pressure or force 
per unit of area. This pressure is of the two sorts already men- 
tioned, viz : first, the passive resistance to external forces which 
is exhibited at its best in the solids ; and, secondly, the sponta- 
neous expansive pressure which is best exhibited in the gases. 
But all known forms of matter exhibit both of these phenomena. 
There are no known solids so dense and hard that they do not 
exert some slight expansive vapor-pressure, although the pas- 
sively resistant form of pressure is overwhelmingly more promi- 
nent. On the other hand, there are no known gases so diffuse 
that they do not exhibit some slight resistance to deformation, 
called 'Viscosit)^," such as is familiar in all liquids and solids, 
although their gaseous characteristics are overwhelmingly more 
prominent. 

Therefore, since both sorts of pressure are to be found in 
finite degree in all cases, and since neither sort of pressure can 
be explained mechanically without finite eccentricity of orbit, it 
must be assumed that all molecular orbits are somewhat eccen- 
tric. Neither circular nor rectilinear orbits are possible. 

This supposition agrees, too, with the mechanical principles 
developed in the Third Paper: That eccentricity of orbit could 
be removed only by radial action, and therefore that, as the 
eccentricity decreased and the radial phenomena became less and 
less, the difficulty of further reducing the eccentricity became 
greater and greater ; so that it is unimaginable that eccentricities 
should ever be reduced to zero, by activities depending upon 
the eccentricity for their effectiveness. Zeros of pressure and 
zeros of eccentricity of orbit must be alike dismissed from con- 
sideration, as conditions impossible of occurrence in nature, con- 
stituting limits which may be approached but never reached. 

The same is true of infinity of eccentricity, or zero of 
curvature, of orbit. Radial departure between two bodies can be 
created only by tang^ential action at periastron, as in the illustra- 
tion of the cannon-ball. Therefore some radial component must 
always be retained. It is imoossible to imagine tangentiallv im- 
parted intensity of "adial motion ever getting to the point where 
there was no tangentiality ; or where, in other words, the orbit 
had ceased to be a hyperbola and had become a straight line. 



MECHANICAL CONCEPTS 109 

Now, of the two sorts of pressure just mentioned, it is plain 
that outwardly active, or expansive, pressure can be explained 
only by orbits possessing eccentricities greater than unity. Only 
in such cases would either member of a mass-pair be able to free 
itself from its mate or mates, and fly outwardly until arrested 
by external resistances. Passively resistant pressure, however, 
which absorbs energy in forceful resistance to deformation, but 
which makes no effort to expand beyond fairly fixed limits, can 
be explained only in terms of orbits having eccentricities below 
unity; that is, elliptic orbits. For all such orbits contain within 
themselves a fixed outward limit of motion, beyond which there 
will be no trespass. But to any arbitrary limitation of motion 
within those limits, by forces exerted from without, to the short- 
ening of the natural length of the ellipse, there would be exerted 
stout resistance. 

Since both sorts of pressure are found in all natural condi- 
tions of matter, it is necessary to assume that in all molecular 
energy-systems there exist both elliptic and hyperbolic orbits. 
Since the passive form of pressure is much greater than the 
expansive sort in solids, it is necessary to assume that in solid 
matter the far greater portion of molecular mass is revolving in 
elliptic orbit, only minor fragments following hyperbolic orbits. 
In gaseous matter, on the other hand, it is necessary to assume 
that the major portion of the mass moves in hyperbolic orbit, 
only a minor portion retaining elliptic motion. 

It is only in the unattainable, limiting case, however, that all 
of the mass could assume hyperbolic orbits ; for it is only by 
action which depends upon mass in elliptic orbit that any other 
mass may be given a superpermanent intensity of energy in 
hyperbolic motion. This fact makes it clear that no combination 
of natural circumstances could ever develop in matter a state 
where all elliptic motion had ceased and all the attributes of a 
solid had disappeared. And this fact, too, agrees with all the 
observations heretofore drawn, viz : That matter and energy 
depart, in either direction, from a central condition where all 
things are balanced, toward extremes where one or the other 
condition becomes exaggerated or suppressed, only with steadily 
increasing difficulty; and that no imaginable conditions or proc- 
esses could ever force things to the point where any of these 
normal attributes of matter had become zero. But such an 



110 ENERGY 

impossible state of affairs as matter in which all orbits were 
hyperbolic and none elliptic would constitute, if it could ever 
be attained, the much talked of ''perfect gas." Therefore the 
''perfect" gas does not and cannot exist. Reliable scientific 
authority does not teach that it does or could; but the general 
concept of the perfect gas has been used so irresponsibly, and 
with such mischievous results, by many teachers of engineering 
thermodynamics, that we shall refer again to its absurdity as a 
concept of natural matter. 

To return now to the concept of expansive pressure as a 
manifestation of hyperbolic motion: Such a concept of pressure 
as a bombardment of the walls surrounding a hot substance by a 
multitude of radiating molecules is by no means new. The 
trouble with it is that it doesn't explain. The trouble is not that 
such a bombardment could not exert the pressure. The trouble 
is that the pressure is exerted continuously, without loss ; whereas 
every bombardment known to human experience involves several 
losses. All the projectiles are lost. All their energy is lost. And 
usually the wall itself is also lost. 

It is of no use, in this juncture, to have it explained to us that 
the wall is perfectly elastic and the projectiles are perfectly 
elastic, and that both wall and projectiles are indestructible. We 
know nothing about any such things. The bombardment-ex- 
planation of pressure has been to the author, ever since he first 
heard it as a student, a blind failure to explain the obscure, 
because attempted with the aid of something still more obscure. 
And, so far as he can discover, it has been equally so to every 
sincere student. 

Another aid, and also obstacle, to the comprehension of pres- 
sure lies in its similarity to and its contrast \^ith temperature. 
If we are to rely upon the linear kinetic energy of the outwardly 
flying particles having hyperbolic orbits to explain pressure, what 
is left to explain temperature? Moreover, temperature has 
already been defined as this linear kinetic energy. For pressure 
and temperature, while sufficiently alike in some respects to be 
considered identical, are 3^et strongly contrasted in some other 
respects. 

Speaking broadly, the active expansive pressure of the vapors 
and gases Is roughly proportional to temperature. Similarly, the 



MECHANICAL CONCEPTS 111 

passive resistant pressure of the solids is inversely proportional 
to temperature. It is the higher temperatures which create vapors 
and gases, with their great expansive and slight resistant pres- 
sure. It is the lower temperatures which develop soHds, exhibit- 
ing the reverse of this. In the more permanent gases the pro- 
portionality of expansive pressure to temperature is almost exact, 
according to Boyle's law, while the viscosity, or passively resist- 
ant pressure, or "solidity" as we might truly call it, is almost 
inversely proportional. In the liquids the vapor-tension varies 
directly and widely, though not exactly proportionally, with the 
temperature. In the solids the connection between expansive 
pressure and temperature is much more obscure, but it is roughly 
visible and it is nowhere reversed. 

All of this evidence, if it were not for the bombardment diffi- 
culty, would fall in excellently with what was said as to elliptic 
and hyperbolic motions. The almost circular, or low-eccentricity, 
motion of the "solid" molecules is fitted as naturally to explain 
the stout passive resistance of the solids as the highly eccentric 
hyperbolic motion of the "gaseous" particles is to explain active, 
expansive pressure. For a given mass can most effectively resist 
a deflecting force (without itself undergoing transformation) 
when it is moving at a high velocity normally to that force ; and 
a circle is normal to every line approaching its center from 
without. The hydraulic jets of the old-fashioned placer-mines, 
in California, for instance, were said to come from their nozzles 
with such a velocity that a man could not strike an ax into the 
water. Therefore it is quite reasonable to imagine the solid 
state of matter as consisting of mass-particles situated very 
close together, yet kept apart by a very high velocity of almost 
circular motion about each other; for in such case the velocity 
must increase as the particles come more closely together, or in 
other w^ords, the hardness and rigidity of passive resistance to 
external pressure must increase with the density — which is just 
what is observed in nature. 

Such a system would be elastic, but non-expansive. The 
abstraction of energy would increase both density and hardness 
of resistant pressure, while the addition of energy would soften 
and expand it into a liquid or a ^as. Everything about the 
explanation would be beautifully consistent, if only some sufficient 
substitute for the bombardment could be found, to explain the 



112 ENERGY 

method of application of these external forces and energies to 
the tiny revolving systems. 

The key to this situation, as also to the contrast between 
pressure and temperature, lies in the following facts. When 
pressure performs work, there is no transfer of energy across 
the gap where the pressure is felt. Whenever and wherever 
temperature is felt, however, there does occur a transfer of 
energy across the gap. 

The first of these statements appears at first sight paradoxical. 
Nevertheless it is true. When pressure performs work it is the 
surface bounding the energy which moves, not the energy across 
the surface. 

To explain, when pressure performs work the energy is most 
commonly what the writer calls transient energy. In transient 
energy the true source of energy is some more or less distant 
driver, from which the energy comes by some carrier which is 
itself a mere inert ''pusher," exerting the pressure which is 
under discussion. In such cases the amount of energy trans- 
mitted is quite independent of the mass of the carrier. Instances 
of transient energy are found in the fluids supplied to hydraulic 
cranes and pneumatic tools, where a remote pump or air- 
compressor furnishes the energy, which the water or air under 
pressure merely transmits to the driven piston. 

The same is true of the steam of a steam-engine, during 
admission. Then the energy is furnished solely by hot-water 
expansion in the boiler ; the steam in the steam-pipe and cylinder 
undergoes no energetic change whatever, but serves merely as a 
fairly frictionless transmitter, neither expanding nor contracting, 
nor undergoing acceleration or retardation. It is only after cut- 
off that the steam itself becomes an active source of energy. 

It is transient energy, too, which is active in the water of a 
penstock. It is only within the guide-blades and wheel that the 
water's own energy becomes active. It is transient energy which 
gives to the greater number of machine-parts their value. All 
rods, shafts, chains, belts, etc., are busy handling transient 
energy. The only pieces of metal in which we utilize the energy 
stored in the metal itself are weights, springs and hammer-heads. 

In all these cases it is obvious that it is the surface bounding 
the energy, the surface where the pressure is felt, which moves 
as the energy is transmitted, and not the energ}'- across the 



MECHANICAL CONCEPTS 113 

surface. In all cases of transient energy the energy-fund of 
each piece remains constant. It is the piece itself which moves. 

In the case of non-transient energy, such as weights, springs, 
hammers or bodies of expanding steam or gas, where the energy- 
fund of the body does change, it is still true that no energy is 
transmitted by pressure across the surface of contact. It is only 
as that surface moves that energy is, or can be, transmitted. 
The fact that the energy is developed within the body, rather 
than transmitted through it, does not alter the case. A weight 
resting at constant height, a spring under constant distortion, a 
hammer-head which hits no anvil, or a body of steam under 
constant volume — none of them transmit any energy, although 
they may exert great pressure. 

It is the essential characteristic of expansive pressure, then, 
if we are to follow the mechanical hypothesis strictly, that it 
consists of a bombardment — because it is associated only with 
the hyperbolic paths of particles which would not turn back 
except for the external pressure — but of a bombardment in 
which no energy is transferred. 

Now such a bombardment is very different from anything 
known in military experience. Such a bombardment would 
require the party bombarded to kindly catch all projectiles, with- 
out impact or friction, and to return them with velocity con- 
served and direction of motion reversed. Such an action has 
been provided, in the explanations of pressure commonly given, 
by the assumption of "perfect elasticity." But the reversal of 
direction of motion with perfect conservation of energy is only 
found in nature, it has repeatedly been pointed out, in the 
mutual revolution of mass-portions around one another. There- 
fore, if the hypothesis that heat is a mode of motion is to 
remain intact, pressure must be regarded as the result of the 
radiation from the surface in question of a continuous flow of 
tiny projectiles; but these projectiles, instead of actually striking 
and rebounding from contact with the particles of the other 
body (for pressure can be felt only betzveen two bodies), must 
be imagined as being met half-way by a similar swarm of pro- 
jectiles radiated from the other body, as revolving about them 
in a "swing-opposite-partners" fashion, returning home with 
pressure exchanged but energy conserved. 

According to this, there can never occur what is commonly 



114 ENERGY 

called "contact" between mass-portions. All action is at a dis- 
tance. Contact, so-called, is merely the approach of two systems 
of particles into such propinquity that they begin to feel — or 
rather, to manifest perceptibly to our crude senses — the repulsive 
effect of the bombardment of projectiles from the other body. 
In gases and vapors these projectiles travel far and embody 
considerable mass; most of the mass of the body is in projectile 
form; therefore the repulsive effect is far reaching. In solids, 
on the other hand, the projectiles are few and of short range. 
The major portion of the mass of each molecule is engaged in 
elliptic motion, exerting no expansive pressure. Such a central 
and self-contained portion of the solid molecule is called the 
nucleus, in contrast with the projectiles which it sends out upon 
hyperbolic orbits. In the case of solids the expansive pressure 
of these projectiles can scarcely be felt. But as soon as the 
external pressure drives these feeble projectiles home upon the 
rapidly revolving (and therefore rigid) nucleus, the resistance of 
the latter promptly becomes considerable. 

An excellent simile for this situation is the exchange of 
pressure between bodies of troops in war. There the sense of 
contact is exchanged by means of flying projectiles, or bullets. 
Each body experiences a force of repulsion from the other by 
means of these projectiles, long before the two bodies come 
visibly into contact. An observer of military action from some 
commanding height, who was familiar with primitive warfare 
but knew nothing of guns and bullets, would be much surprised 
and puzzled in seeing two orderly masses of men, moving in 
opposing directions with antagonistic aims, slow down as they 
approached within a considerable distance of each other, hesitate, 
waver, desist from their predetermined purpose, break into small 
and irregular detachments at the points of closest approach, take 
on lateral motions of various sorts, and finally the smaller body 
reverse its direction of motion in response to the repulsion of the 
larger, and depart with a velocity higher than that of approach — 
all of which is quite similar to what we observe in the ''contact" 
between solid portions of matter. The imaginary observer of 
all this could not see the flying bullets which were the active 
agents in this transfer of pressure from army to army, and he 
would be entirely at a loss to explain it, no matter how wide 
had been his experience with warfare conducted with swords, 



MECHANICAL CONCEPTS 115 

spears and chariots — until some one mentioned the bullets. 

Now the impulse carried by the bullets is a moral one, rather 
than physical, and is dependent as much upon the accuracy and 
timehness of their discharge as upon their mass and velocity. 
So this simile is truly a simile, rather than an accurate scientific 
analogy. Yet it may be extended usefully to the contrasts 
between gases and solids. An army progressing freely through 
territory where opposition is not imminent, divided into many 
separate bodies, with extended columns, flying detachments of 
cavalry, scouting and foraging parties, skirmish-lines, etc., may 
be likened to a gas. It will occupy all the space it can. 

Such an army, meeting another such, would feel the pressure 
of contact very gradually. A skirmish-line would be driven 
back here or there; scouting parties and flying detachments 
would become more careful and stay nearer home. But only 
after appreciable time would these outlying features be driven 
into consolidation with the main body, and the entire mass be 
compressed into a form so dense that pressure felt at any one 
point would be transmitted with promptness throughout the entire 
mass. 

The armies thus condensed, as in active conflict in difficult 
country, where the enemy's pulse is not easily felt, might be 
likened to solids. In such case there would be little premonition 
of contact, until it occurred with considerable force. Then there 
would be little elasticity. Instead of repulsion in visible reverse- 
motion or diversion of path, the exchange of energy would be 
sharp and sudden, and would occur with great loss. Contact 
would become synonymous with impact. The balance of power 
would be established through grinding wear, with the partial 
destruction of the reacting bodies, in disintegration and heat of 
conflict, rather than with a graceful yielding to the pressure from 
without and a gradual accumulation of available mass against 
the point of contact, as is the case with elastic bodies. 

In all these ways the likeness between the contact of bodies 
of troops and that between bodies of physical mass in motion is 
very great. In both cases what is called contact proves, upon 
careful examination, to be no contact at all. It is merely an 
approach into sufficient propinquity so that a perceptible force 
of repulsion is experienced. And wherever the word perceptible 
is used it refers as much to the thing doing the perceiving as it 



116 ENERGY 

does to the thing perceived. There is no end of things in nature 
which are imperceptible to our fairly crude human senses, when 
applied directly, which have been made plain either by the greater 
sensitiveness of inanimate instruments or by scientific analysis. 
And apparently the universality of the manifestation of repulsive 
pressure throughout the universe, between each two bodies, is 
one of these. 

The further development of this idea of the transmission of 
pressure by projectiles, as the only possible explanation of ther- 
mal phenomena which is in accord with Newtonian mechanics, 
will be deferred for a continuation of this chapter. 



CHAPTER X. 

The Mechanical Concept of Pressure (Continued). 

The mental picture which must be formed of the average 
molecule, in order to prosecute the mechanical analysis of heat, 
is as complex and varied as the aspect of an army-corps in 
active campaign — to complete the simile which was introduced 
in the preceding paper. It must itself consist of many, very 
many separate portions. 

The major portion of this mass, in the solids and liquids, at 
any rate, will be devoted to the formation of a central body 
which may be called the nucleus. This nucleus is itself complex 
in structure, formed of many parts, which may separate, upon 
occasion, in independent action. Its internal motions are more 
or less mobile and fluid. Yet it embodies a greater degree of 
consistency, and approaches more nearly to the condition of a 
solid, than the more out-lying portions. In the mechanical con- 
cept of heat this nucleus must be conceived as composed of 
particles revolving in elliptic orbits, many of them of an eccen- 
tricity approaching zero. In this way the nucleus may be 
understood to possess a unified existence of its own, of fairly 
permanent dimensions, until its internal equilibrium is upset 
and its unity broken up, by its invasion with sufficient force by 
some disturbing mass from without. 

Surrounding this nucleus is a swarm of particles possessing 
hyperbolic orbits. These, to distinguish their activities from that 
of the nucleus, will be called the satellites. Nevertheless, it is 
not necessary to assign any other distinguishing feature to the 
satellites than their high eccentricity of orbit. They may be 
regarded as capable of becoming absorbed at any time by the 
nucleus, or of being formed from nuclear particles. Indeed, it 
is natural to assume that this process of transfer of particles, 
from nucleus to satellitic swarm and the reverse, as one particle 
or another gains or loses the intensity of energy requisite, is 
going on at all times. This may occur as readily as, in the 

117 



118 ENERGY 

similar army-corps, men are constantly being detached from 
head-quarters for assignment to distant posts of duty ; while 
other men, these duties done, are as continually returning, for 
re-absorption by the central body. 

Nor is it necessary to imagine any sharp definition between 
the nuclear swarm and the satellitic swarm. The nucleus, on the 
one hand, may possess some members which, although moving in 
almost circular orbits, yet lie far out away from the main mass. 
Such instances occur in our own solar system, for instance, in 
such planets as Neptune and Uranus, which have not completed 
one of their "years" since the dawn of modern astronomy; or 
in one of the moons of Jupiter, which is so far distant from 
that planet that it is barely caught permanently, against the sun's 
attraction. 

The satellitic swarm, on the other hand, may possess many 
members whose great eccentricity of orbit carries them as near 
to the center of mass at periastron as they are distant at apastron. 
Such particles, which would correspond to the comets of our 
solar system, would alternately dive into the very center of the 
nuclear swarm, and then depart as remotely into space as 
external mass-systems might permit. The instance, in our solar 
system, of the Great Comet of 1882 has already been mentioned. 
Here is a vast mass which last experienced its maximum separa- 
tion from the mass-center of the solar system when Columbus 
was struggling into manhood. At that time the comet was at 
aphelion, far beyond the orbit of the most remote known planet, 
drifting lazily in parallel with the planetary orbits. But whereas 
the planets possess sufficient tangential motion to prevent them 
from falling toward the sun, this comet did not. It began to fall 
before Columbus set sail for the Indies, and for four hundred 
years thereafter it fell continually, with velocity increasing each 
second, through unknown millions of miles, toward the sun. 
Finally, in September, 1882, it reached its mean energetic con- 
dition, very near the sun. Within ninety minutes thereafter it 
had further transformed into kinetic form an equal amount of 
energy with that transformed during the preceding four cen- 
turies, and had reached its extreme energetic condition, at peri- 
helion. Within a second period of ninety minutes this gigantic 
scale of energy-transformation had been reversed, the second 
mean energetic condition had been passed, and the comet was 



MECHANICAL CONCEPTS 119 

away upon another eight centuries of outward and return motion, 
almost purely radial in character. 

Yet a molecular particle acting similarly to this must be 
classed as belonging to the "solid" nucleus, rather than to the 
expansive-pressure-exerting satellitic swarm, because its orbit is 
elliptic. The influence of no external mass-system is needed to 
return this satellite to the sun. No external mass-system would 
feel its pressure until it had encroached upon definite dimensions 
of our solar system. 

Therefore the molecular mechanical system is to be imagined 
as consisting of all sizes and velocities of mass-portion, moving 
in all sorts of orbits. Some will move in almost circular orbits 
of small radius and very high velocity, others in the same form 
of orbit with large radii and low velocities. Some will move 
in ellipses of high eccentricity, with large and small mean ener^ 
getic distances. Some will move in hyperbolic orbits, tending to 
lose themselves from the system except as they are returned to 
it by outside forces. Indeed, the mechanical idea of the molecule 
must also include the constant loss of some of these smaller 
particles from the molecule, their mass being made up, under 
average conditions, by an equal absorption of freely roving par- 
ticles lost by other molecular nuclei — just as an army-corps both 
loses and gains men continually, in the form of veterans disabled 
home and raw recruits received, and in the form of prisoners lost, 
taken and exchanged with the enemy. 

In a general way, the proportion of satellitic energy to that 
embodied in the elliptic orbits of the nucleus is given by the 
proportion of "external work" visible in the substance in question. 
Thus, in the case of steam, a glance at the steam-tables will 
show that the proportion of any increment of heat going respect- 
ively to internal and external work differs widely in ice, water, 
superheated steam and the mixture of dissociated hydrogen and 
oxygen. If we remember that the "external" work is that por- 
tion of the energy which is devoted to holding at bay the sur- 
rounding bodies which press in upon the water, steam, etc., it 
is plain that the work involved is that stored in the satellitic 
swarm. During the flight of each satellite its -fund of energy is 
stored in kinetic form. During the "swing-opposite-partners" 
reversal of motion at the far end of the route, where the pro- 
'jectile comes into "contact" with the external mass, such as a 



120 ENERGY 

boiler-shell or engine-piston, which is being pressed back by the 
hot, elastic body, this kinetic energy takes the form of * 'pro- 
pinquity," or intensity of spacial approach. Immediately there- 
after the motion is reversed, toward the original nucleus, and 
the energy becomes kinetic again. 

Therefore, the prominence of external work in the energetic 
action of any body is an excellent criterion of the proportion of 
the mass of each of its molecules which is engaged in satellitic, 
or hyperbolic, orbital motion, as contrasted with that which is 
involved in nuclear, or elliptic, orbital motion. 

In general, both the nucleus and the satellites will tend to 
arrange themselves in a generally spherical form. While, no 
doubt, an individual variation of form away from the spherical 
could be detected in each molecule, could it be examined closely, 
yet the general tendency of all particles would be to dispose 
their departures from the center equally in all directions, and to 
about equal distances. 

So soon as two molecules should approach within "per- 
ceptible" propinquity, the form of each of their internal orbits, 
and the general form of each molecule, would be altered, by the 
attraction of the mass of the other molecule. The situation may 
be illustrated by Fig. 9, wherein the several typical forms of 
orbits, and their perturbation by the mutual approach of the two 
molecular nuclei, A and B, is shown by dotted orbits marked 
a, h, c and d. 

Many of the orbits, including most of those embodied in the 
nucleus, which before were almost circular, would experience, as 
the result of this attraction, merely a lengthening of their major 
axes toward the other molecule. These are shown at a and h. 
Other orbits, such as those shown intersecting at c, which before 
had been ellipses much elongated toward the other molecule, 
would find their major axes shortened at one end, the end nearer 
the other molecule, by the mutual attraction between the particles 
themselves and their mutual revolution about each other at c, in a 
little periastron of their own. Indeed, in the greater number of 
such cases the orbits would before have been the hyperbolic ones 
of satellites exerting expansive pressure outwardly ; but now they 
are drawn down into ellipticity by their own mutual attraction. 
Such orbits, however, would not be geometrically perfect ellipses. 



MECHANICAL CONCEPTS 121 

Other orbits, such as d, are those which, originally hyperbolic 
in reference to A or B alone, are now drawn down into ellip- 
ticity by the combined attraction of both, but perform alternate 
periastrons about first one and then the other. Nor are these 
orbits necessarily elliptic. They may be of the form of a figure 
eight, or looped in still more intricate form, when many more 
than two interacting molecular nuclei are involved. The entire 
process of interchange of pressure, indeed, may be likened to 











/ \ --^ _^^ 



FIG. 9. 



some of the figures of the old-fashioned quadrille, in which "right 
and left" or "ladies' chain" or "dos a dos" furnish opportunity 
for sturdy nuclear gentlemen to set delicate satellitic ladies flit- 
ting about orbits of the most intricate and confusing character. 
In which connection it is of further interest to note that all of 
these old dance-forms, like most savage dance-figures, arose from 
the primitive custom of amusement with mimic warfare, as in 
the joust or tournament, in which advance and retreat, skirmish 
and charge en bloc, were so arranged as to exchange pressure, 
but maintain equilibrium, between two opposing parties. 

Any one of these satellites, at every point in its orbit, is 
subject to two attractive forces, one drawing it toward the 
nucleus A and the other toward B. In each case the attraction 
reacts upon the nucleus with the same force that it exerts upon 
the satellite. The two attractive forces, however, are seldom 



122 ENERGY 

equal. At any such point as e, for instance, the two forces 
would bear the ratio 

F eB 

assuming that the masses of A and B are equal. But this same 
satellite, if its orbit were of the flf-class (or some chance equiva- 
lent in the horde of satellites if it were not), would also be 
found later at the point /, engaged in periastron, around the 
other nucleus B. The point / being symmetrically opposite to Cy 
the forces developed at / will be equal and opposite to those at e. 
Only now the larger force is exerted upon the nucleus B instead 
of upon A. 

The resultants of all of these different forces acting upon the 
two nuclei A and B will always tend to alinement with the axis 
AB joining the two bodies, and will always be separative, or 
repulsive, in its direction. The more nearly the two bodies 
approach, the greater will be the proportion of the mass of each 
which engages in the sort of modified orbit which develops these 
repulsive forces. 

If the two systems, each consisting of nucleus and swarm of 
satellites, should be generally circular or spherical in outline 
(according to whether confined or not to a single plane), as is 
our solar system, and should be of uniform density, these 
repulsive forces would vary inversely as the square of the 
distance between the nuclei. While there is no need for forcing 
any particular assumption as to the form of the molecule, this 
much is said to show that any ordinary laws of variation of 
force with propinquity may be explained in a sensible fashion, in 
terms of such a mechanical system. In general, it may be said 
that any two such systems, upon approach, would first be elon- 
gated in the line of their approach, w^hich would increase their 
radial susceptibility, or elasticity, in that direction, and would 
afterward be mutually repelled, in a perfectly elastic manner, by 
forces increasing with some power of the propinquity. 

Early in this series of papers it was shown that true elas- 
ticity can be conceived, in consistency with the Newtonian 
mechanics, only when the relative motion of a mass-pair is 
reversed by the circumrevoluiion of the two members. It now 
becomes clear how perfect elasticity may be manifested between 
two complex ma.ss-systems by the circumrevolution of only a 



MECHANICAL CONCEPTS 123 

portion of each mass-system about a similar portion of the 
other. The proportion of the total mass which is thus neces- 
sarily involved in penetration into and circumrevolution about 
the other depends, of course, only upon the intensity of the 
relative motion between the systems ; that is to say, upon the 
amount of V- in proportion to the M^ -f- M2 present. It is hence 
easy to infer how it is that intensity of energy is a controlling 
factor in determining energy-transformation; for the penetra- 
tion of more than a certain proportion of each system into the 
other would break up the internal equilibrium of both, resulting 
in either their amalgamation or their permanent rupture ; in short, 
in their ''transformation." 

In imagining any such systems as those of Fig. 9 as con- 
stituting actual molecules, it is to be remembered that the pro- 
portion of satellites to nucleus may be widely variable. In the 
case of solids the nuclei must embody the greater portion of the 
mass of the molecule. They must be small in diameter, very 
dense, and very rigid from their high speed of rotation. The 
satellites to correspond would be few, but would possess great 
density and high velocity. In the case of gases, the nuclei would 
possess large diameters, low densities and relatively low periph- 
eral speeds. The satellites would be relatively of much greater 
mass than in solids. 

If the proportion of satellitic to total mass of the molecule 
be represented by k, then the repulsive force developed by two 
approaching systems would vary, not only inversely with some 
power of the distance, but proportionally to M^(k — k^). This 
expression reaches its maximum when k=:J or when the mass 
is equally divided between nucleus and satellites. It is therefore 
to be expected that the possibilities for elastic force of repulsion 
should be the greatest in some intermediate stage, such as the 
liquid, where the satellites would bear a medium proportion to 
nucleus, rather than in the extreme conditions of solid or gas, 
where either nuclear or satellitic mass preponderates. This 
seems to be true in nature. In the gases the elasticity is well- 
nigh perfect, but the force of repulsion is small. In the solids 
the force of repulsion is very great, but the elasticity is quite 
imperfect. In the liquids, however, the force resisting com- 
pression is very great, and at the same time the elasticity is 
excellent. 



124 ENERGY 

Should any pair of mass-systems such as Fig. 9 be forced, by 
great energy and directness of impact, into closer propinquity 
than would permit the nuclei to remain intact, they will become 
broken up. They may then separate in fragments, or coalesce 
into a new nucleus, of larger dimensions and of the same or 
different form of arrangement. It is thus that the processes of 
welding solids, mixing liquids and gases, and impact and friction 
can be explained. Chemical combination may also be imagined 
as explained in the same way, although the premises must then 
be stated in a much more intricate manner. Obversely the 
accumulation within any one molecule of too much intensity of 
energy might easily lead to its bursting or splitting into two or 
more separate molecules, of the same or of different chemical 
nature.* 

Another result imaginable from the too close collision of 
nucleus with nucleus would be the production of more satellites, 
from the fragments of the ruptured nuclei. The importance of 
this natural mechanical action, as an explanation of heat-forma- 
tion by impact, was pointed out in the Seventh Paper, on "What 
is Heat?" It is now to be pointed out, in addition, that this 
process, like all other true energetic processes, develops a con- 
dition of stable equilibrium. That is to say, the impact of the 
nuclei was due to a paucity of satellites. The result of the 
collision is to reduce the nuclei partially to satellites, thus making 
good the deficit of satellites and diminishing the chances of 
another similar collision's occurring. 

The beautifully stable balance thus established pervades all 
nature. It has already been referred to briefly, in our primary 
definition of heat. Its wide results will be taken up later at 
length, in the chapter upon Thermal Equilibrium (Chapter XV). 

In discussing the action of these colliding systems it is hardly 
necessary to point out that the distance-factor, for instance, must 
possess a vastly different scale in mechanical, celestial and thermal 
energetics, respectively. Distances which constitute mean ener- 
getic ones here on the earth's surface, in the applied mechanics 
of engineering, would constitute prodigious extremes of separa- 
tion when measured between molecules, but corresponding 



*The mechanics of this hypothesis has been beautifully developed in 
the theory of the evolution of stellar nebulae into dumb-bell form, with 
their ultimate consolidation into "double" stars. 



MECHANICAL CONCEPTS 125 

extremes of propinquity, or concentration of matter, when meas- 
ured between planets. 

It is necessary to point out, however, that a similar range in 
the mean energetic values of the other factors of energy, such as 
force, velocity, density, etc., is to be expected as one passes from 
mechanical to celestial energetics, on the one hand, or to molec- 
ular energetics on the other. Yet there is no reason to suppose, 
from these differences in degree, that the principle of action is 
at all different. For instance, there has been every reason to 
suppose that the principles of mechanics were applicable to the 
explanation of thermal phenomena. And yet, if so, how are the 
enormous forces which are needed to account for thermal ener- 
gies to be explained? Thus, one pound of carbon, in the form 
of coal, by virtue of its separation from 2f pounds of oxygen 
in the atmosphere, embodies in the total mass of 3f pounds 
some 14,600 B.t.u. or 11,360,000 foot-pounds of potential energy. 
All of this is released kinetically upon letting the carbon and 
oxygen fall together, or "burn." Equating this quantity to the 
equation for potential energy, in which the quantity-factor M^Mg 

would equal X — -"= 0.00258, it results that So, the 

32.16 32.16 

closest distance of approach between carbon and oxygen, must 
be only about 10"^ inches. But this is much closer than the mean 
diameter of molecules, as estimated from other sources, has 
been accepted as being. According to Professor J. J. Thomson, 
the diameter of the molecule is about 4 X io~^ inches, while that 

of the electron is about 4 X lo'^"^ inches.* 

From this the conclusion is driven inevitably to one of tv/o 
things. First, the carbon and oxygen must each be subdivided 
into particles much smaller than a single molecule, which so pene- 
trate the other's molecular swarms as to attain a mean pro- 
pinquity far greater than that permitted by molecular diam- 
eters. Secondly, the density of mass in the molecular particles 
is far greater than that of matter as we are familiar with it in 
the solar system. Probably both statements are true. 

This is the equivalent of saying that what has hitherto been 
considered and measured in diameter as a molecule is not to be 
considered as a solid homogeneous sphere, but as a multiplex 

*Engineering (London), March 19, 1909, page 391. 



126 ENERGY 

mass-system, containing much space as well as material particles. 
It was Professor Rowland, we believe, who said, many years 
ago, that *'a grand piano, in comparison with a molecule, is sim- 
plicity itself." As to what may be the proportions between 
space and matter in the molecule, nothing may be said with 
confidence. Pursuing, however, the plan hitherto followed, of 
deducing all our molecular ideas from celestial mechanics, the 
molecule should follow the proportions of our solar system. 

But in speaking of densities, in connection with solar systems 
and molecules, it is too commonly assumed that the densities of 
individual planets, such as our earth, constitute our sole and 
proper guide to the densities of molecular matter. But if mole- 
cules are to be likened to solar systems, a collection of molecules, 
such as ordinary matter, must be likened to a collection of solar 
systems. The density of the particles composing the molecule 
must bear the same proportion to the density of the entire mole- 
cule, or to a collection of molecules, as the density of planetary 
mass bears to the mean density of a solar system or a galaxy. 

What, then, is the mean density of our solar system ? Assum- 
ing that the orbit of the most distant planet, Neptune, constitutes 
the outer limit of the solar system — although many comets con- 
trolled by our sun far exceed these limits — and remembering 
that a molecule is a system occupying space of three dimensions, 
whereas the orbits of the planets occupy virtually only space of 
two dimensions — the mean density of the space occupied by the 
solar system is easily computed as being less than that of our 
solid earth by about the ratio lo^^. This means that the mean 
density of such a system, when perceived from far enough 
without so that it appears as a unit, is about that of one three- 
hundred-millionth of an atmosphere, or far rarer than the matter 
within an electric-light bulb, or even a Crookes tube. If these 
same proportions are to apply to a molecule of carbon dioxid, for 
instance, then the mean density of the smallest particles which 
engage in heat-motion, and which will retain the characteristics 
of carbon and oxygen, respectively, must be some million millions 
of times greater than that of our solid earth. 

There is no intention here to place any weight upon the exact 
arithmetical aspect of these proportions. All that is intended is 
that, if our ideas of heat-energy are to be explained in terms of 
mechanics, and if our mechanics are to be drawn from the only 



MECHANICAL CONCEPTS 127 

known source of true mechanics — celestial mechanics — then there 
appears to be no difficulty in explaining the heat-energy stored 
chemically in the dissociation of elements, as due to simple 
gravitation between mass-particles. It is just as probable that 
the density of such related particles is many millions of times 
greater than the densities of matter familiar to human discern- 
ment as it is that molecular dimensions and distances should be 
many millions of times smaller. It is the wide range in densities 
thus opened before us, as we deal with more and more finely 
subdivided mass, which makes it possible to explain even the 
energy of radium, which embodies a million times the energy of 
an equal mass of oxygen and hydrogen, in terms of mechanical 
energetics. Similarly, the explanation of the heat-energy stored 
in the physical disgregation of matter in vaporization or fusion 
is still easier, for the amount of energy stored per unit of aggre- 
gate mass is much less. If it should happen that not all of the 
vast range of dimensions which the above figures show to be 
open to the imagination is necessary for the explanation, so 
much the better. If it should happen, too, that the dimensions, 
densities, etc., which are requisite for the mechanical explanation 
of heat do not immediately fit the estimates which have been 
drawn from other considerations, the discrepancy is for the 
physicists to explain. The doctrines of the Newtonian mechanics, 
and of heat as a mode of motion and separation within mass, are 
both of them now too broadly founded in scientific experience 
for them to bend easily to meet other empiricisms which may be 
discrepant therewith.* 

A second intention of these statements of enormous ratios is 
to impress upon the mind the fact that the solid forms of matter 
with which the mechanics of engineering is chiefly concerned 

*The writer wishes to repeat and emphasize here the caution which 
appears in the preface, that these statements are not to be regarded by the 
student as coming from one who has investigated molecular activities at 
length in the physical laboratory. They are founded merely upon the 
premises which have been frequently repeated throughout the work, as 
appertaining to the whole — the Newtonian mechanics and the doctrine that 
heat and work are one. But, in oral discussion of these matters with those 
who are entitled to speak as p'hysicists, he has frequently met the objection 
stated, that it was impossible to explain the gigantic forces and energies 
of molecular mass mechanically. But in every case, upon investigation, It 
has developed that this position has been based upon some quite unwar- 
ranted and limiting assumption — such as this that molecular mass must b" 
of densities similar to mundane density — which is in reality gratuitou'^ 
and unnatural. It is to show that such assumptions are not only need- 
less, but groundless, that this argument has been inserted. 



128 ENERGY 

constitute, when viewed in proportion to the linear distances and 
volumes also employed, a most extreme condition of concentra- 
tion of matter. If these engineering bodies were to be located 
properly on the field of Fig. 8, or Fig. 12, for instance, their 
place would be found at the extreme prolongation of the curve 
CB beyond B. That is to say, iron, steel and granite, as used in 
machine construction, are about as far removed from their 
natural mean energetic conditions as ice, when at a temperature 
of — 400° Fahr., is removed from that condition of boiling water 
or superheated steam wherein it displays its greatest thermo- 
dynamic adaptability for handling large quantities of energy, 
both in thermogy and labority — the condition in which it explodes 
our boilers, drives our engines and ameliorates the sudden 
changes of climate and season. 

It now appears that a finally exact definition of heat is a 
very difficult thing. In the Seventh Paper it was defined as the 
spacial and kinetic relativities between the particles of a body, 
excluding the subpermanent or colliding particles. Now it seems 
that the definition of even so simple a term as collision is difficult. 
Collision is an approach so close as to upset the permanence of 
equilibrium of molecular existence. A few years ago we should 
have said that this settled it: We knew what permanence meant. 
Now questions are raised as to the permanence of existence of 
chemical matter, by the degradation of radioactive matter, which 
are hard to answer; and heat can scarcely be a more permanent 
form of energy than chemical energy. 

Moreover, little light has yet been shed upon the question 
whether the superpermanent, or hyperbolic, energies of the 
molecule should be included as heat. To the writer, they plainly 
should. 

It will be found of great assistance to the understanding of 
heat-action, in the next paper, even if we do not know just 
what heat is, to have done these things, viz : ( i ) To have dis- 
posed of the idea of the "perfectly elastic yet solid" molecule; 
(2) to have reduced the definition of contact and collision to 
their proper places; (3) to have similarly disen throned the 
usurper called "perfect gas"; and (4) to have established heat- 
action similarly with mechanical action upon the basis of the 
mean energetic condition, depending upon action at a distance, 
rather than upon that natural unreality, contact. 



CHAPTER XL 

The Two Basic Thermal Processes, 
heat-transfer and work-performance. 

Chapter IX announced the need, in thermodynamic discussion, 
of mechanical concepts of pressure, volume, temperature and 
entropy. The first two of these obscure properties of matter 
have now been discussed, in their mechanical aspect. Before 
proceeding to a similar discussion of temperature and entropy, 
however, it will be necessary to develop the mechanical concepts 
of the two basic thermodynamic processes, heat-transfer and 
work-performance, as contrasted with static thermal attributes, 
such as pressure, volume, temperature and entropy. 

The thermal diagram, Fig. 8, which was presented in Chapter 
VIII (page lOo), developed before the eye only two out of the 
several simple thermal processes which are familiar to boiler 
and engine-room. These two were, first, the isomorphic and 
metathermal"^ heating and cooling of bodies. This process was 
explained as if it were performed only by heat developed by 
impact and friction ; but heat supplied by conduction or radiation 
would have been found to produce identical results. The second 
process was the metamorphic and isothermal one of addition or 
abstraction of heat when the body was neither heated nor 
cooled, but changed its physical state instead, as in fusion or 
vaporization. This too was explained with heat furnished by 
impact or friction, although heat produced by conduction or 
radiation would have produced the same results. In the latter 
case, however, the original form of the energy would have been 
as obscure to the understanding as the final one (whereas we 
feel that we understand the mechanical nature of impact or 
friction) and so would have been of little aid to the under- 
standing. 

The familiar thermal phenomena of the power-house, hov/- 
ever, include at least two other processes which need explanation. 

*Meaning "temperature-changing," contrasted with isothermal. "Meta" 
signifies change, as "iso" signifies constancy. 

129 



130 ENERGY 

These are, first, adiabatic work-performance by heat, and, sec- 
ondly, the isenergic degradation of heat by wire-drawing. It will 
be assumed that the reader is sufficiently familiar with both of 
these processes to avoid the necessity of their general descrip- 
tion. However, in order to make prominent the features which 
are sufficiently characteristic to aid in understanding their internal 
mechanical operation, they will be briefly defined. 

The Adiabatic Process. Adiabatic action occurs only when 
a body undergoes simultaneous change in volume and tem- 
perature, while expanding or being compressed, with the exclusion 
of heat-interchange by conduction or radiation or friction or 
impact with other bodies. The only permissible exchange of 
energy with the outside world is in the form of work; and this 
exchange is unavoidable, if the volume is to alter, for pressure 
exists everywhere. 

The energy thus exchanged is supplied from the body's fund 
of heat; and not only from its fund of heat, but from its fund 
of temperature-he2Lt. For no other sort of heat is available for 
work-performance. Not only does adiabatic work-performance 
always result in temperature-change, but it is the only knozvn 
exact measure of temperature-change. This fact is the basis of 
Carnot's foundation of thermodynamics, and of Lord Kelvin's 
perfect scale of temperature, in which equal "degrees" are defined 
in terms of equal amounts of work performed, not in terms of 
equal amounts of heat transferred ; nor, as in our ordinary ther- 
mometry, in terms of equal increments in volume of expansion. 

In adiabatic action the alterations in volume and temperature 
occur in inverse directions. The temperature decreases as the 
volume increases, and vice versa. The process is represented in 
the thermal diagram, Fig. 8, by a straight vertical line. 

The Wire-drawing Process. When a fluid, whether gas- 
eous or liquid, flows through a pipe or orifice, the flow is resisted 
by friction. The thermodynamic effect of this takes either one 
of two forms. 

I. If the pressure upon the substance be greater than its 
vapor-tension, for all the temperatures in question, the tem- 
perature of the fluid will rise and its pressure fall as it proceeds. 
This process is an isomorphic one of heating by friction and 
impact, as already fully described, except that it occurs under 



TWO BASIC THERMAL PROCESSES 131 

falling pressure. Such is the action, for instance, in all water- 
pipes of ordinary temperature. 

2. If the pressure at any point should become less than the 
vapor-tension corresponding to the local temperature (and it is 
obvious that Process No. i tends to result in this state of affairs, 
if carried far enough), then the wire-drawing process ceases to 
be isomorphic and becomes metamorphic. The substance begins 
to vaporize. Such, for instance, is the action in the ordinary 
boiler-blow-off, particularly if the blow-off pipe be imagined as 
so long that virtually all of the energy of the escaping fluids is 
spent in overcoming friction. The same action occurs in the 
expansion-valve of the ammonia refrigerating-machine. 

Or, if the substance happens to be already vaporized when it 
starts upon its impeded flow, as is the case with steam flowing 
through a steam-pipe, then the result of the isenergic action will 
be to further rarefy it. This action is familiar in the ordinary 
throttling and superheating calorimeter. 

Between these two numerated forms of isenergetic wire- 
drawing there exists a most instructing similarity and contrast. 
The similarity lies in the fact that both processes increase the 
quantity-factor of the heat, or its entropy. The contrast lies in 
the fact that whereas the first process mcreases the temperature, 
the second c?^creases it. 

The conclusions to be drawn are obvious, and are two-fold. 

1. Friction and impact are intimately and functionally linked 
with entropy-change. 

2. Frictional or impactive increase of entropy occurs inde- 
pendently of, and has no determinative effect upon, temperature. 
The question as to whether the temperature is to rise or fall or 
remain constant, as the result of friction or impact, depends 
solely upon the external pressure. For, as has just been pointed 
out, the increase of entropy by impact or friction under rising, 
falling or stably constant temperature are processes equally 
familiar in nature. 

In order to be sure of understanding this position, it were 
well to examine further the nature of wire-drawing. 

Taking, for convenience, the second, or expansive, form of 
wire-drawing, it soon reveals itself as a duplex process. It con- 
sists, in reality, of two distinct processes merged into an appar- 
ent, but not real, unity. First comes the acceleration of the mass 



132 ENERGY 

of liquid or gas into linear motion. This absorbs work; and as 
the only source of energy for this purpose is the body's heat, the 
temperature falls to supply it. This part of the wire-drawing 
process is purely adiabatic. 

But the linear motion of the particles is no more than gene- 
rated than it engenders friction ; and the word friction is merely 
our short name, as has been seen, for the destruction of linear 
motion in the increase of the quantity-factor of heat. In the 
present case, since the rate of flow is in equilibrium with the 
friction opposing it, the rate of energy-transformation, all around, 
is controlled by the degree of friction present. The rough, hard 
walls turn the work of flow into heat as fast as they can, and 
the consequent deficit in mechanical energy is made good con- 
tinuously by the further adiabatic expansion of the body. 

Now it happens that wire-drawing is our only instance of 
simultaneous adiabatic expansion and friction. The steam- 
turbine seems at first to offer another and more illustrative case. 
But upon second thought it appears that its internal action is the 
same as pure wire-drawing, except that it is complicated by the 
simultaneous presence of still a third process, viz : adiabatic 
work-performance upon outside systems. In steam-engine cyl- 
inders we have an obverse illustration, viz : the addition or 
abstraction of heat while adiabatic work-performance is going 
on. But to bring this into line with pure wire-drawing we must 
introduce the inference, which is everywhere forced upon us, 
that friction and thermal conduction are identical processes, in 
their ultimate nature ; except that whereas conduction will work 
both ways — in and out — friction will work only one. 

Nevertheless, these facts serve only to broaden, rather than 
to narrow, the conclusions stated above, viz : 

1. That friction affects only entropy, being indeterminative 
of temperature-change ; 

2. That the direction of temperature-change depends solely 
upon whether there be a surplus of external over internal pres- 
sure, radially active, or not ; and 

3. That in the apparently single process of wire-drawing 
there really exists a merger of both the above independent proc- 
esses, viz : a variation of the entropy by friction, and a variation 
of the temperature by pressure-action or work-performance. 

If we turn now from the development of heat by friction or 



TWO BASIC THERMAL PROCESSES 133 

impact to its twin brother, the development of heat by thermal 
conduction, there appears a most striking similarity to the wire- 
drawing situation. That is to say, not only does thermal con- 
duction identify itself, by ways not necessarily repeated here, 
with friction and impact as an inevitable controller of entropy- 
change, but the temperature-change tvhich accompanies it is not 
determined by the quantity of heat conducted, but solely by the 
relation between external and internal pressures. If the sub- 
stance be water, for instance, and if the pressure upon it be 
greater than its vapor-tension, then the addition of heat by 
thermal conduction, or by friction either, increases the tem- 
perature — or rather, it results in an increase in temperature. 
But if the external pressure and the vapor-tension be balanced, 
as is the case in the ordinary steam-boiler, then the addition of 
heat by thermal conduction, or by friction either (though in this 
case we lack a familiar instance), results in no rise in tem- 
perature. The water merely vaporizes isothermally. 

If, again, the external pressure happens to be less than the 
vapor-tension — as is the case in a steam-engine cylinder w^hen 
the resistance of the piston is less than the expansive force of 
the steam — then the addition of heat by thermal conduction, 
such as occurs if the engine-cylinder be steam-jacketed, occurs 
in connection with a fall of temperature. Or the wire-drawing 
of steam, already described, is an even better illustration. 

We have finally come to the point, therefore, where the 
absurdity of calling temperature-change the effect of either 
friction or thermal conduction has become obvious. It is entropy- 
change alone which is this. Temperature-change must be the 
functional result of something quite different. In cylinder- 
expansion we need no assurance that it is the deficit of pressure, 
with the consequent retreat of the confining walls, which creates 
the drop in temperature. The simultaneous addition of heat has 
nothing to do with the case, except in the alteration of the 
entropy present. But in the addition of heat to the water in a 
boiler, thermal conduction has no more to do with the tem- 
perature-n>^ than it has, in the steam- jacketed cylinder, with the 
the temperature-/a//. It is pressure alone which controls both. 

It is only because the great majority of all the things to 
which we have been accustomed to add heat have had vapor- 
tensions below that of the surrounding atmosphere, or other 



134 ENERGY 

external pressure, that we have come to associate the addition 
of energy by thermal conduction, or by friction and impact, with 
rise of temperature. This popular association of these two 
phenomena is not only loose and inaccurate, but it is funda- 
mentally erroneous and misleading. It begets in the student's 
mind an idea of interdependence between temperature-change 
and thermal conduction which is in direct antithesis to the 
principle laid down by Lord Kelvin, viz : that the only correct 
measure of temperature-change, the only correct thermometric 
degree, is zvork-performance. It draws the very foundation 
from beneath what should be the first and fundamental idea 
taught the student in the thermal laboratory, viz : that thermal 
conduction has everything to do with alterations in entropy, and 
nothing whatever to do with alterations in temperature. 

The deliberate association of thermometry and thermal con- 
duction in the physical laboratory should cease. The first efforts 
of the instructor in thermodynamics, after a clear concept of the 
general nature of energy is once implanted, should be directed 
toward a breaking up of this false association of ideas, already 
too deeply imbedded by the ordinary experiences of life before 
reaching the study of the natural sciences. 

ONLY TWO FUNDAMENTAL THERMODYNAMIC PROCESSES. 

The conclusion is inevitable, therefore, that there exist but 
two basic, elementary, thermodynamic processes. One of these 
is the generation of thermal quantity-factor, or entropy, by 
impact, friction, thermal conduction or any other process which 
proves upon examination to be an equivalent: a process which 
always occurs at constant temperature, or independently of and 
ineffectively upon the temperature. 

The other process is the adiabatic variation of temperature 
by work-performance : a process which always occurs at constant 
entropy, or independently of and ineffectively upon entropy- 
change, but which cannot occur independently of temperature- 
change. 

In order that the fundamental importance of these two proc- 
esses may be clearly brought out, they may be given distinctive 
names and completely defined, as follows : 

I. Thermogy: the isothermal variation of entropy, occur- 
ring always when matter is subjected to impact, friction, radia- 



TWO BASIC THERMAL PROCESSES 135 

tion, thermal or electrical conduction, or perhaps other processes 
as yet unidentified. In the mathematical discussion of thermogy 
the temperature T must always appear as an integer, and the 
entropy-change dN as a differential. 

2. Labority: the isentropic, or adiabatic, variation of tem- 
perature, occurring always when matter develops work from 
heat, and sometimes when matter develops heat from work — that 
is, when it absorbs work radially with perfect elasticity, or 
"reversibly." In the mathematical discussion of labority the tem- 
perature-factor dT must always appear as a differential, and the 
entropy- factor N as an integer. 

The first of these processes must always be represented in 
the thermal diagram. Fig. 8, by a straight horizontal line, prop- 
erly of infinitesimal length, but integrable, under proper conditions, 
into a finite extent. The second process must be represented by 
a straight vertical line, under similar conditions. 

All other thermodynamic processes whatever, of which several 
are commonly depicted upon the thermal diagram by various 
oblique and curved lines, must be considered as made up, in 
natural fact as well as geometrically, of these two basic com- 
ponents, occurring simultaneously and combining to produce an 
apparently single, though really double, result. 

Thus, in the isomorphic heating of water under pressure, 
which is represented by the oblique curve ADW, Fig. 8, the 
energy is supplied in such form (either by friction, impact, con- 
duction or radiation) that the quantity-factor is increased by the 
value dN, say. The quantity of energy thus supplied, dQ, is 
equal to T dN and is represented by a vertical element of area 
between ADW and OX. The result of this supply of energy is 
to vaporize momentarily a minute quantity of the water. 

If the external pressure upon the water be exactly balanced 
with the vapor-tension, as along DE, this minute portion of 
vapor remains vapor; and further increments of vapor, from 
further increments of heat, also remain vapor, cumulatively. 
But if the external pressure be in excess, as is necessary for 
the prosecution of the isomorphic AD toward D', the tiny bits 
of vapor are recompressed, adiabatically, into water, as fast as 
they are produced, leading to the adiabatic rise in temperature 
dT. The result of this simultaneous increase of quantity-factor 
and temperature is portrayed by the curve DW. 



136 ENERGY 

The other isomorphic curves might be similarly explained, 
except that in the case of ice the reaction cannot occur between 
solid and liquid forms, because pressure turns ice to water, not 
water to ice. The only way out of the difficulty is to assume 
that there must be present in the ice, at all temperatures, minute 
quantities of both water and water-vapor, between which the 
reaction occurs ; and this fact has been developed independently 
by direct observation. In the case of refractory solids, which 
refuse to fuse under all ordinary applications of heat, it is to be 
remarked that the resistance to fusion is plainly involved in the 
crystalline form of the solids — a question too complex and 
bearing too lightly upon power-engineering for discussion here. 

Throughout the previous discussion the quantity-factor of 
heat has appeared as a matter of fundamental import. Obvi- 
ously, it also needs a short and suitable name. Hitherto it has 
been referred to often merely as "the quantity-factor," in order 
that the discussion might not be vitiated by doubts as to the 
propriety of a name, or by preconceived notions as to the nature 
of entropy, the only name suitable for it. But the analysis has 
now reached a stage where these considerations need no longer 
defer the identification of the quantity-factor of Fig. 8 with the 
term entropy which was invented by Professor Clausius fort}^ 
years ago. Fig. 8 itself may be similarly identified with the 
entropy-temperature diagram of Gibbs and Macfarlane Gray. 

It should be explained immediately that this identification is 

not complete nor exact nor final; but this is so only because 

doubt exists as to the natural facts upon which the definition 

of entropy rests. Professor Clausius defined his understandino^ 

of entropy as being a function "equal to or greater than, but 

dO " . 
never less than, the integration of — — — in any thermodynamic 

cycle. The equality was to apply when the cycle is free from 
impact and friction, or was ''reversible." The increase in entropy 
would occur when these were present — as they always are. 

But Clausius's definition of entropy, while deserving the very 
great credit of being the first to appear, is nevertheless incom- 
plete. It is incomplete in regarding itself as complete and final. 
It is incomplete in three distinct ways. 

First, while Clausius's definition of entropy is exact enough 
in terms od dQ, he gives no definition of dQ. None of the text- 



TWO BASIC THERMAL PROCESSES 137 

books which have followed Clausius attempt to define it. We 
are still to-day, although possessing a vastly greater accumulation 
of data than in Clausius's time, unable to define dQ. Clausius 
defined dQ in terms of impact, friction, radiation and conduction. 
We know now that that list is incomplete ; but what to add to it 
(beyond some types of chemical, electrical and magnetic action), 
and how we are to know when we have finished the list, we are 
yet unable to say. 

Secondly, Clausius recognized no other form of entropy than 
thermal entropy. In Clausius's day, although the Conservation 
of Energy had been foreshadowed by Newton (1680) and 
defined and pictured pretty completely in the writings of Mohr 
(1837, Mayer (1843), Grove (1846) and Helmholz (1847), 
two decades before Clausius, yet the idea that heat is a mode 
of motion was still obscure in the scientific world. It was far 
beyond the natural, for that day, for Clausius to see that when 
he had identified entropy-production with impact and friction, 
which are purely mechanical, he had proven that there must be a 
mechanical form of entropy, even if there were none of heat or 
temperature. Thus, long before the brilliant work which began 
with Count Rumford and ended with Kelvin, Joule and Maxwell 
had settled beyond peradventure the physical reality and the 
mechanical nature of both heat and temperature, the real (albeit 
undeveloped) foundation for the identification of entropy also, as 
a physical reality and a feature of mechanical energy, had been 
laid, in the original discovery of the function itself. 

To-day the doubt lies not as to whether entropy be a physical 
reality, such as temperature, heat or work, rather than a mere 
mathematical function. It does not even lie in the direction of 
whether there may be other sorts of entropy than thermal entropy. 
Our difficulty to-day is to know where the concept of entropy, as 
a physical reality, is to end. We know that as far as energy- 
transformations extend, so far do all of the forms of energy 
exhibit the dual attributes of intensity and extensity, or quantity- 
factor, respectively. If the energy itself is identical throughout 
all these changes, so must be the two component factors. There 
must be as many forms of temperature and entropy as there are 
forms of energy interchangeable with heat. But for their exact 
definition we must await further knowledge. 

Thirdly, Clausius saw no chance for the entropy of any 



138 ENERGY 

system ever to decrease. Knowing no other form of entropy 
than thermal entropy, this was most natural ; for thermal entropy, 
so long as it stays thermal, does tend constantly to increase — 
upon the earth's surface, at any rate. But more, so far as 
scientific thought may surmise, everywhere in nature exists some 
solidity, or at least viscosity; and wherever solidity or viscosity 
exists, mechanical motion is being constantly converted into heat 
and heat is degrading itself in temperture, with increase of 
entropy. 

Starting from these facts Professor Zeuner announced his 
theorem that the entropy of the universe tends always to a 
maximum. Proceeding further from this same base, Lord 
Kelvin gave countenance to the cosmic doctrine of the general 
degradation of all forms of energy into unavailability ; and the 
school of what is now orthodox thermodynamics was settled 
beyond discussion.* Whether there be such a thing or not as a 
universal degradation of energy will be discussed in a later 
chapter. The writer may announce here that he does not believe 
that there is. He believes that the availability of the energy of 
the universe remains constant, although in regions as great as a 
solar system it may locally and temporarily fall into deficit or 
accumulate into a surplus. 

But it is not upon any principle so broad as this that the 
concept of entropy must rest. It rests upon the familiar, every- 
day transformations of energy, and upon the fact that therein 

*The following extract is taken from a lecture by Sir Oliver Lodge 
upon Lord Kelvin, which was delivered after these pages had been written : 

"I fancy that he himself, and certainly some of his disciples, have 
been at times inclined to attribute to the law of degradation mort ulti- 
mate and cosmic importance than properly belongs to it. Its significance 
is limited to the validity of the terms 'heat' and 'temperature' ; and 
if for any reason these cease to have a practical meaning, then the dissi- 
pation of energy ceases to be inevitable. * * The different availabilities 
of various kinds must be essentially a human and temporary conception, 
useful and convenient for practical purposes, but not ultimate nor 
cosmic. * * The dissipation of energy has no meining when 'heat' 
and 'temperature' are obsolete terms ; that is to say, when what we now 
consider to be unorganized and intractable molecular motions can be dealt 
with in an individual and organized way." 

Professor Zeuner, too, is to be credited with the first perception of the 
physical reality of entropy. His term for it, "heat-weight," is a most 
graphic expression of one of the prime characteristics of entropy, namely, 
its "heft" or forcefulness in work-performance. But entropy, we shall 
see, is not the force of thermal gravitation, but the degree of subdivision 
of matter which develops that force, and which, taken with propinquity, 
determines its magnitude. 



TWO BASIC THERMAL PROCESSES 139 

appears everywhere this duality of intensity-factor and quantity- 
factor of energy, running through all the natural sciences. Its 
importance is basic, not only in the cases of heat and work, 
which alone can be discussed here, but in chemical, electrical, 
magnetic and radiant energies as well. Although it is only in 
mechanical and thermal energies that the quantity-factor has 
been accurately defined — as S (M^Mg) or the extent of mass- 
pairing present in the first, and as entropy in the second — yet in 
all the other sciences its existence and importance are none the 
less indubitable because vague. In all these fields it demands 
immediate definition. Not only must the engineer, the electrician 
and the chemist contribute to the physicist their data, before this 
general energy-factor which in heat finds the name entropy can 
be properly defined, but the biologist, the economist and the 
sociologist must follow suit. In all of these sciences there has 
been too much time devoted to study and measurement of the 
visible space-and-motion factors, and far too little attention given 
to understanding what it is which can experience space or 
motion. There need be no hesitancy in stating that when our 
students are properly taught the nature of entropy, in its broadest 
sense, they will find the concept of equal use, in after life, in 
their science, their engineering, their business and their politics.* 
For all these reasons the writer will use the term entropy in 
a considerably wider, and a somewhat different, sense from that 
given it originally by Professor Clausius (if a bare and rigid 
mathematical definition can be called a "sense"). He will use 
it as synonymous with the quantity-factor of heat, and he will 
show in the next paper that it is identical with the quantity- 
factor in mechanical energy, S (M^Mg). He will also suggest 
briefly the idea that this same quantity-factor runs through all 
the other forms of energetic action to which the laws of the 
transformation and conservation of energy apply, in the inani- 
mate, vegetable, animal human and social activities of the uni- 
verse. It will appear that no student may lay claim to a grasp 
of the fundamental principles which guide and control any of 
these activities, without first acquiring a thorough comprehension 
of the mass-factor in mechanical energy — that it is the form of 

*The reader who desires a further develooment of this idea will 
find its discussion in the Harvard Encinerrin^ Journal for 1008, J^muary 
and November issues. The broader significance of the quantity-factor of 
energy receives further discussion in the last chapter of this book. 



140 ENERGY 

association, or grouping, or relationship, between things, and not 
the things themselves, which gives rise to energy, and to the 
characteristics of its results. 

The next thing is to understand this corresponding thing in 
heat which we call entropy. The final fruit is to comprehend 
that all forms of energetic activity, including social energetics, 
are based upon the same factors and follow the same laws as 
those which support and guide the silent stars, the incandescent 
flame and the awful thunderbolt. 



CHAPTER XII. 

Mechanical Concepts of Thermal Phenomena. 
B. temperature and entropy. 

The definitions given in the preceding paper for the two 
basic thermodynamic processes, thermogy and labority, now pave 
the way for clear mechanical concepts of temperature and 
entropy. Before attempting to formulate any such concepts, 
however, it were well to get well in mind the peculiarities which 
are characteristic of the two quantities respectively. 

Temperature. The prime characteristic of temperature, and 
the only true measure of temperature-change, is work-perform- 
ance. Work can be performed at the expense of heat only 
when the temperature falls. Work can be absorbed elastically, 
with obvious conservation of energy, or "reversibly" as it is 
called, only when the temperature rises. 

This idea of temperature we owe originally to Carnot (1825), 
who stated the law that the work performed — if the process 
were perfectly elastic, or ''radial," or ''reversible" — was pro- 
portional to the temperature-fall ; that is, the law is conditioned 
upon pure labority, devoid of thermogy — the straight, vertical, 
adiabatic process. Half a century later this idea was amplified 
by Lord Kelvin into the principle that the only accurate measure 
of temperature-variation, or the only true thermometric scale, 
was to be based upon equal degrees, not of heat-supply, nor of 
volume of expansion, but of work-performance. 

The second characteristic of temperature is the universal 
tendency of differences of temperature to set up energy-transfer 
by thermal conduction and radiation. This is the explanation of 
the familiar sensations of hot or cold — the gain or loss of 
energy by the skin and nerves when brought into contact with 
other bodies, or when exposed to radiation. These ideas of 
temperature long antedated the discovery of the work-scale of 
temperature ; yet they are by no means such accurate guides in 
the understanding of the nature of heat. They must be con- 

141 



142 ENERGY 

sidered quite secondary to the prime characteristic of tem- 
perature-heat : work-performance. 

These sensations are deceptive because they concern merely 
the rate of transfer of energy, a thing which is subject to other 
influences than mere temperature-difference. Thus, exposure to 
the wind, or wetting the skin, will heighten the sensation of 
cold. The latter of these two processes may actually lower the 
temperature and be perceived by the thermometer; but the former 
makes no difference to the thermometer. 

A still more accidental characteristic of temperature is its 
effect upon the volume of a substance. All gases and vapors 
increase in volume with rising temperature. Some liquids and 
solids do the same, though at a lower rate ; but rnany reverse the 
rule and increase in volume with falling temperature. Yet this 
characteristic of temperature is our main reliance in ordinary 
thermometry. The relative expansion of mercury and glass is 
the most common means for exhibiting temperature-changes. 
For more accurate purposes hydrogen is substituted for the 
mercury. 

A similar characteristic of temperature is the proportionality 
to it of the expansive pressure of gases and vapors. This 
proportionality, like that of volume, is never strictly true, even 
for gases, and for saturated vapors it is quite approximate. In 
solids and cold liquids the vapor-tension becomes an insignificant 
affair. While neither pressure nor volume is ever exactly pro- 
portional to the temperature, the very fact that the departure 
from proportionality is least and almost zero in the gases, and 
is greatest in the solids, is itself a guide to the understanding 
of temperature — as will be developed later. 

Entropy. The prime characteristic of entropy is its indiffer- 
ence to reversible or elastic work- performance, by or upon the 
body, with accompanying temperature-change. Its secondary 
characteristic is its sensitiveness to and inseparability from 
impact, friction, conduction and radiation. Its third charac- 
teristic is its general proportionality with volume. While the 
relations between entropy and volume are even less regular than 
those between pressure or volume and temperature, yet they 
are obvious. They stand out most plainly in the process of 
vaporization, wherein the increase in both volume and entropy is 
large, and is exactly proportional. 



TEMPERATURE AND ENTROPY 143 

The last characteristic of entropy to be mentioned, but not 
the least in importance, is its general proportionality to elas- 
ticity. This, while inexact, like volume, is yet seldom reversed. 
i\n increase in entropy almost always means an increase in 
elasticity. It is this fact which maintains thermodynamic hap- 
penings in an equilibrium which is almost always stable. Impact 
and friction are features of inelasticity. Impact and friction 
always develop entropy. Entropy is always accompanied by 
elasticity. Elasticity reduces impact and friction. Thus, the 
phenomena v/hich are due to a deficit of elasticity result in the 
cancellation of that deficit and a retardation of the phenomena. 

Labority and Thermogy. The facts needed to complete 
the data for the mechanical definition of temperature and 
entropy are those of the two basic thermodynamic processes 
already defined in the preceding paper, viz : labority and ther- 
mogy. Labority, or work-performance, is identified with tem- 
perature-change, but does not affect entropy. Thermogy is 
identified with entropy-change, but does not itself affect tem- 
perature. 

Radial and Tangential Action. It is next to be noted that 
work-performance by heat is always accomplished by an elastic 
movement of the surface of the hot body normally to itself ; that 
is to say, radially in reference to the molecules composing the 
surface-layer. And if the expansion of the body is supposed to 
be effected by a simultaneous expansion of all of its molecules, 
this too would obviously be a radial movement on the part of 
each one. 

Thermogy, on the other hand, consists of an inelastic move- 
ment of the surface of the hot body in its own plane; that is to 
say, tangentially in reference to the molecules of the surface- 
layer. In the case of rubbing friction, this fact is obvious. In 
the case of direct impact it is less obvious. But even here it is 
only necessary to note that impact consists, by its very definition, 
of the energy which is not returned radially or elastically, to see 
that impactive energy must also be tangential in its mode of 
transfer. 

As to conduction and radiation, all that can be said is that 
nothing positive is known as to the form of their action, but 
that their results are identical with those of impact and friction. 
Until positive evidence to the contrary arises, therefore, it is 



144 ENERGY 

natural to assume that impact, friction, conduction, radiation, and 
even electrical resistance, are all identical, as to their form of 
molecular transfer ; that is to say, that they are all tangential, and 
not radial, activities. 

It is now to be recalled that in the study of the elementary 
mechanical mass-pair there was found a marked contrast between 
the radial and tangential funds of purely mechanical energy. 
The radial fund was easily defined with mathematical accu- 
racy. It was the perceptible fund of energy. It constituted the 
medium of communication with outside mass-systems. Its 
intensity was what determined whether the energy should undergo 
transformation or not : kinetic intensity above the critical point 
producing dissociation; spacial intensity (of concentration, or 

propinquity, or—-) above the critical point producing collision. 

The tangential energy-fund of the pair, on the other hand, 
was found to be incapable of exact definition, although alterations 
in it could be measured exactly. It was the imperceptible or 
latent fund of energy. It was self-contained within the mass- 
pair. It carried on no direct communication with outside sys- 
tems. Nevertheless, it was capable of receiving or sending ener- 
getic messages to the outside world, through the medium of the 
radial energy, to an indefinite extent. 

Mechanical Concepts of Temperature, Entropy and 
Heat. The data and mechanism for the comprehension of ther- 
modynamic happenings, as activities of mass, motion and space, 
are now before the eye, fairly complete. 

Heat consists in both motion and space between the molecules 
of the hot body. Each molecule is itself a complex system of 
particles, possessing motion and space relatively to each other; 
and these internal relationships are also heat, in part at least. We 
are not attempting to define the molecule itself, nor the atoms 
which form parts of a molecule, nor the ions or electrons which 
may form parts of the atoms. That is for the chemists and 
physicists to do. All that is being said here is that, if heat is to 
be regarded as a mode of mechanical motion at all, it must be 
regarded as a very complex system of motions and spacial 
separations, some of them between the molecules, and some of 
them within the molecules. In the gases the motion is chiefly 
between the molecules, each molecule moving bodily along hyper- 



TEMPERATURE AND ENTROPY 145 

bolic paths which are almost straight Hnes ; but some of the heat- 
energy still remains within each molecule. In the solids the 
motion is chiefly within each molecule, but there is still some 
motion between each two molecules, of tiny projectile particles 
if not of the whole molecule. 

These relative motions are of two sorts, or components, viz : 
radial and tangential. Each orbit contains some of each. The 
elliptical orbits have more tangential component than radial, the 
hyperbolic orbits more radial than tangential. 

Each molecule contains a nucleus of particles moving in 
elliptic orbit, and a swarm of satellites moving in hyperbolic 
orbit. In the solids the nucleus is the major feature, as to mass, 
and the satellites are the minor; in the gases the satellites are 
the major and the nucleus the minor. In fact, vaporization may 
consist in the nucleus becoming satellitic. In the so-called ''per- 
fect," or absolute, gas the entire mass would be satellitic. In the 
absolute solid the entire mass would be nuclear. Neither condi- 
tion is ever attained in nature. 

Volume. Each tiny mass-pair embodies a certain degree of 
spacial or radial separation, which vibrates constantly above and 
below its "mean energetic distance." In the mean energetic con- 
dition the gravitational or centripetal attraction is balanced 
against the centrifugal force of tangential motion. The mean 
energetic distance of separation at which this equilibrium is found 
determines the volume of the hot body. 

In the solids the mean energetic distances are very small and 
the tangential velocities very high. In the gases the distances are 
very large (comparatively speaking) and the tangential velocities 
low. For at larger radii smaller tangential velocities are sufficient 
to maintain equilibrium. 

The radial velocities, on the other hand, vary in the reverse 
order. In the solids they are low and in the gases they are high. 

Temperature and Pressure: In this intricate swarm of 
particles of all sizes, moving in orbits of all dimensions, forms 
and velocities, the net or integrated effect of all the radial activi- 
ties is both temperature and pressure. The integrated intensity 
of energy is temperature. The integrated momentum is pressure. 

Matter, even in its most quiescent states, finds itself always 
in contact with other matter, in solid, liquid or gaseous condi- 
tion. At the point of contact both temperature and pressure are 



146 ENERGY 

exerted. If a temperature-difference exists at the point of con- 
tact, energy is transmitted across tlie gap between the bodies, in 
the form of heat; but the gap does not move. If a pressure- 
difference exists at the point of contact, no energy is transmitted 
across the gap ; but the gap itself moves, and energy is imparted 
in the form of work. 

In the first case the energy transmitted comes from the in- 
ternal, or molecular, energy of the matter immediately adjacent 
to the point of contact. In the second case the energy trans- 
mitted comes either very slightly or not at all from this con- 
tiguous matter. In the case of ''transient" energy it comes from 
some more or less distant point, and reaches the point of contact 
solely as the pressure-energy of a mass which itself moves 
bodily. In the case of energy developed adiabatically by the 
expansion of the body in contact itself, it originates all over the 
body, between each nucleus and its satellites, simultaneously ; and 
is then transmitted "transiently" to the particular satellites en- 
gaged in "contact." 

When temperature-difference exists at the point of contact, 
the two bodies may or may not exchange iemperature. That 
depends entirely upon the pressures prevailing. The only thing 
certain is that they always exchange entropy. A pure illustration 
— that is, one where the process is not complicated by the influ- 
ence of pressure-changes — is the multiple-effect evaporator, in a 
sugar-works. There heat is transmitted from steam condensing 
under a higher constant pressure and temperature, on one side 
of a metallic wall, to water evaporating under a lower constant 
pressure and temperature on the other side. The wall serves to 
neutralize the pressure-difference, but performs no thermal office. 
The two steam-bodies exchange only entropy. Neither under- 
goes temperature-change. 

But they can exchange entropy only when a temperature- 
difference exists. The office of temperature is apparently what 
might be called catalytic : to effect the transfer of entropy, with- 
out being itself affected. 

Unfortunately, all of our ideas concerning heat are so instinct- 
ively founded upon the sense of hotness to the touch, as the 
prime criterion of thermal conduction, that this simple phenom- 
enon is hard to understand. We say that a substance is "hot," 
when we touch it, because we find that the finger has been 



TEMPERATURE AND ENTROPY 147 

heated by it. But if the finger be not heated, any amount of 
heat might be transmitted to it, from a substance of much 
higher temperature,, yet we should not say that the latter was 
''hot" — except by inference. Thus, the steam-hot water in a 
boiler, for instance, would not say, could it speak, when a white- 
hot oil-flame were set against the boiler-shell, that the flame was 
hot; for the water would not be scorched thereby, nor even 
raised in temperature at all ; nor affected in any way except to 
be changed into steam no hotter than the water was before. 

Similarly, when one touches a hot flat-iron with wetted finger, 
the same situation, with its lack of sensation of heat, is 
approached. We judge that the iron is hot, not because it burns 
the finger, but because it makes a "siss" of steam. We feel no 
hotness. We merely hear the rapid formation of entropy and 
volume. 

Now this little experiment gives the only true idea of thermal 
conduction, not as a thing which reveals temperature, but as a 
thmg transmitting entropy — which last we cannot perceive at all, 
except by its volumetric effects in steam-making. The fact is 
most difficult to grasp, because all our lives we have grown accus- 
tomed to telling whether things were hot or cold by trying if the 
finger were heated or cooled in touching them. Nor have our 
physical laboratories, where Lord Kelvin's ideas are well known, 
done all which it seems they might to teach their students better. 

Yet the truth of the matter is that the only situation where 
temperature can really be felt is that, for instance, of the steam- 
engine piston. This piston might, and ought to, be of the same 
temperature as the steam pressing against it. There would then 
be no transfer of heat between them. The pislon would not be 
heated. Yet here alone temperature could be felt. The piston, 
because of the superior temperature of the expansive steam (as 
compared with that of the same substance on its opposite face), 
might be moved, could it speak, to say : "I am propelled; there- 
fore I am impelled to note that this steam on one side of me is 
hot!'' For, if Carnot and Kelvin knew anything about the 
matter, it is work-performance alone, and not heat-transfer, 
which is the only true sign of the presence of temperature. 

With our ideas thus clarified it may be accepted that the 
transfer of heat, by the tiny radiating satellitic particles of a hot 
body, need not be associated at all with the development of tern- 



148 ENERGY 

perature in the body affected. Therefore the puzzle as to the 
mechanical explanation of pressure, temperature and entropy 
resolves itself into the following simple array of conditions : 

1. Temperature has already been identified, in the case of 
gaseous matter, by the mechanical theory of heat, as the kinetic 
energy of the active particles. 

2. Temperature is manifested solely by elastic work-per- 
formance; and work-performance always occurs normally to the 
surface of the body, or radially as to the molecules. 

3. Heat-transfer, by thermal conduction, has proven to be 
identical in its effects with impact and friction; and impact and 
friction have already been identified with melasticity, or molecular 
action tangential to the molecule. 

4. Heat-transfer takes place without motion of the envelop 
demarking the bodies between which conduction occurs ; and the 
only way in which satellites flying along conic-section orbits 
might transfer energy from one swarm to another, zvithout 
motion of the surface bounding the swarm normally to itself, is 
tangentially. 

5. Pressure, however, which is accompanied by no transfer 
of energy across the bounding surfaces, might be exerted radially 
by the flying particles, under the conditions stated. 

It is from this basis that emanated the statement, given above, 
that both temperature and pressure are manifested by the radial 
component only of the motion of the satellitic particles. Tem- 
perature is their integrated intensity of energy. Pressure is their 
integrated momentum. Temperature-heat is the radial compo- 
nent only of the radial energy, which was diefined for the 
elementary mass-system on page 40, 

Thermogy. But, further, it can be added that thermal con- 
duction, like impact and friction, is the tangential transfer of 
energy from the satellites of one swarm to those of another. 
This can be accomplished only by means of the tangential motion 
remaining at or near the apastron tip of the orbit. Even when a 
solid steel boiler-shell, for instance, the molecules of which must 
embody chiefly tangential motion, is heated by a white-hot gas, 
the molecules of which must embody mostly radial motion, the 
energy transferred must be regarded as only the tangential com- 
ponent of the gaseous particles. Because the latter possess but 
slight tangential motion (although plenty of radial energy) there 



TEMPERATURE AND ENTROPY 149 

prevails but a low rate of heat-transfer. The highly eccentric 
orbits of the gaseous particles, beautifully adapted for elastic 
work-performance, are very poorly adapted for thermal con- 
duction. Low rates of heat-transfer, per unit of surface and 
difference of temperature, are broadly characteristic of gaseous 
substances. Neither the gases, the satellites of which are many, 
but of too highly eccentric orbit, nor the solids, the satellites in 
which possess chiefly tangential motion, but are too few in num- 
ber and small in mass, transmit heat well by contact. It is the 
liquids, the satellites in which present the most powerful com- 
bination of both numbers or mass (entropy) and tangentiality of 
motion, which are the best thermal conductors. 

But this tangential transfer of energy at the apastron tip of 
the satellitic orbit cannot occur unless there exists a difference of 
temperature, or radial intensity, between the two swarms. Two 
bodies manifesting the same temperature, as well as the same 
pressure, must have equality in both mass and velocity of satellites. 
But if the temperature of body A be higher than that of B, while 
their pressures remain equal (that is, if momentums remain equal 
while kinetic energies are higher in A), then the mass of A's 
satellites must be smaller and their velocity higher than in B. 
But the satellites of each swarm approach the other swarm in 
all directions, chiefly oblique. Radially their directions and 
momentums are opposed, and neutralize each other. But tan- 
gentially many of them may coincide. In that case the particle 
of more rapid motion, as of A, will overtake tangentially and 
impart em^.rgy to the more slowly moving particle of the colder 
body B. 

Labority. Elastic work-performance by temperature-heat, 
on the other hand, is just what these white-hot gases are best 
fitted for; for that is a purely radial action. This fact is seen 
in the thermodynamic superiority of the gas-engine over the 
steam-engine. When a body performs work by heat, the radially 
flying particles find themselves exerting pressure against (that is, 
revolving about, at the remote end of their orbits) the satellites 
of molecular nuclei which are retreating. They are like tennis- 
balls thrown at the rear end of a retreating freight-train, or a jet 
of water impinging against a retreating Pelton-wheel vane. Their 
direction of motion is reversed ; but they return with their radial 
velocity much reduced. 



150 ENERGY 

Their tangential velocity, however, remains unimpaired; for, 
if the process has been a pure one, there has been no heat- 
transfer. Their orbits are therefore reduced in eccentricity and 
increased in mean energetic distance. The molecule has lost its 
temperature and pressure, but gained in volume. It has lost its 
ability to impress itself upon outside systems, but it has not lost 
its entropy, or the mass-factor or quantity-factor of its heat, 
which gives heft to its energy. It retains to the full its tangential 
components. Its entropy remains constant. It is no longer 
valuable for work-performance, but it is still most efficient for 
heating-purposes. 

Since it will shortly be stated that this quantity- factor or 
entropy is merely the number and mass of the energy-pairs at 
work, it is to be noted here that there is nothing about this 
process of work-performance which should lead to any con- 
solidation of mass, or diminution in the number of mass-pairs 
at work. Work-performance, whether done mechanically or 
thermodynamically, always tends to occur at constancy of mass- 
pairing. Only interference by collision makes it otherwise. 

The obverse of this process, adiabatic compression, is not so 
obviously explained. It is plain that the approach of each two 
molecules, enforced by outside power, must decrease their vol- 
ume. It is easy to see that the increased momentum of the 
greater number of projectiles then met and reversed, per unit of 
area, must increase the pressure. But it is not so clear that this 
must also increase the radial activity, or temperature. 

This will be plain, however, when it is remembered that the 
forces which act upon any satellite at periastron are the result, 
not only of mass, but of propinquity. When the molecules are 
compressed each satellite finds itself forced into greater simulta- 
neous propinquity, at periastron, with a greater number of nuclear 
masses. Its periastron velocity therefore increases ; and this 
velocity, although tangential in direction, has already been 
pointed out as being largely true radial energy. 

There also occurs the direct effect upon the radial velocity of 
the approach of the external masses about which the satellites 
are reversed, just as in expansion. Owing to the approach of 
these masses, like a freight-train or Pel ton wheel being backed 
up against the projectiles which strike it, these are returned with 
increased radial velocities. 



TEMPERATURE AND ENTROPY 151 

Temperature. Temperature, then, is radial kinetic energy. 
Work-performance by temperature-drop is in no way different 
from the performance of work by the velocity-reduction of 
mechanical mass, as in a turbine or Pelton water-wheel. Tem- 
perature-increase by compression is in no way different from the 
acceleration of any mass by a supply of energy, as in the cen- 
trifugal pump or marine propeller. The engineer who can under- 
stand these mechanical devices can understand adiabatic thermo- 
dynamics. 

The doctrine that temperature is kinetic energy is not new, at 
all. What these papers are intended to emphasize, in connection 
with it, is that the kinetic energy which constitutes temperature is 

1. The radial component only of the molecular energy; 

2. It is embodied in mass-particles very much smaller than 
the entire molecule, as well as in the latter itself ; 

3. These facts apply to liquids and solids as well as to gases ; 

4. The ''rebound" of the flying particles cannot be imagined 
as that of perfect elasticity in collision, but must be accepted as 
consisting of circumrevolution without contact. 

Entropy. Turning now to the question of the mechanical 
nature of entropy, and of its generative process, thermogy, it 
might be inferred that these were to be found, because of their 
contrast with temperature and labority, in the tangential motion 
of the particles. They are, indeed, closely connected therewith. 
Thermogy is the acceleration of the particles in their tangential 
component. But entropy is the result of this process, not the 
process itself ; and the natural result of an increase in tangential 
energies is a separation or scattering or subdivision of the total 
mass into a greater number of mass-pairs capable of embodying 
temperature. 

Entropy is what corresponds to the quantity factor M1M2 of 
the elementary mechanical mass-pair. It is, more accurately, the 
2 (MM) of any mechanically energetic system. It is the X of 
Fig. 7, a function of n the number of portions into which the 
total mass is subdivided. It is the quantity-factor N of the 
thermal diagram. Fig. 8. It is the degree of subdivision and 
dispersion of the nuclei A and B of Fig. 9 into satellitic swarms. 
It is, in short, the degree of subdivision, specialization and 
organization of an originally comparatively unified, solid, rigid, 
inert and inelastic mass, into a system capable of embodying that 



152 ENERGY 

space and motion which are to us the sole visible manifestations 
of energy and elasticity. This quantity-factor is always itself 
imperceptible; that is to say, directly imperceptible; but it is 
what gives ''heft" and power to that which alone is directly 
perceptible, viz : space and radial motion. 

Intramolecular Equilibrium. It is the stability of equilib- 
rium within the molecule which necessitates always a readjust- 
ment of either sort of energy, radial or tangential, whenever any 
alteration is made in the other. Thus, in heating ice or water 
isomorphically, only thermogy is performed directly. The altera- 
tions of entropy occur at constant temperature; the subdivision 
of the molecules results from speeding them up tangentially, as 
a potter does his wheel, and not from any radial acceleration or 
rise in temperature. But the immediate indirect effect of this 
thermogy, in the face of a superior external pressure upon the 
molecule, is to upset its internal equilibrium; so that the energ}^ 
which was imparted tangentially, expanding the molecule as a 
fly-ball governor expands by acceleration, is squeezed out radially, 
by the recompression of the molecule adiabatically by the external 
pressure, into an increase in radial components which we call 
temperature-rise. The ice or water gets warmer as it is pounded 
or heated ; but this occurs only because ( i ) it has first been so 
subdivided, isothermally, by the thermogy, as to have become more 
elastic and susceptible of compression; and (2) because the 
surplus pressure requisite for the recompression of this greater 
elasticity is present. If the surplus pressure be not present the 
temperature will not rise. It is the adiabatic recompression of 
the elastic portion of the molecule, by this apparently static 
external pressure, and not the thermogic addition of energy to 
its inelastic portion by pounding or heating, which raises its 
temperature. 

On the other hand, when thermogic impact or friction or 
heat-conduction leads to the melting of the ice or the vaporiza- 
tion of the water, without incidental rise in temperature, it is 
because the accumulation of internal radial intensity has become 
sufficient to counterbalance the static external pressure. The 
equilibrium has become indifferent. The expansion of the mole- 
cule by thermogic increase of its tangential velocity now jumps it 
into a new condition of stable equilibrium, that of saturated steam 
— like a fly-ball governor breaking some of its resisting links — 



TEMPERATURE AND ENTROPY 153 

and in that condition it remains stably. No reactive compression 
can occur to raise the temperature, and we therefore say that 
vaporization occurs isothermally. 

In any such a thermogic action the chief absorbent of energy 
would be the separation, or disgregation, of the particles. It was 
pointed out in Chapter I that the expression for space-energy 

was proportional to -^^ ^, and that in this expression the value 

of So had much more to do with the value of the function than 
did that of S. In other words, when mass is separated the work 
involved depends much more upon the original degree of con- 
centration, or density, of the mass than upon the final degree of 
diffusion. This work of separation is called disgregation-zvork 
— the word disgregation signifying the ''scattering of a flock," or 
the opposite of congregation. This disgregation work is very 
great for solids and liquids, because of their comparatively great 
density. They are unusual congregations or concentrations of 
mass, embodying unusual deficits of energy. For gases it is much 
less, decreasing to almost zero as the diffusion becomes great. 
But because mass can never be so widely separated that its 
mutually attractive force becomes zero, so no gas can ever 
become so rarefied or "perfect" that its disgregation-work in 
change of volume ever becomes zero. 

In illustration of this mechanical concept of the thermal mole- 
cule, the writer has used the simile of a juggler standing before 
a table carrying a stock of balls. The total stock of balls repre- 
sents the molecule. The portion which the juggler succeeds in 
keeping in motion in the air is the satellitic portion. The 
remainder upon the table is the nucleus. In this simile the 
nucleus would have no motion at all, whereas in the actual 
molecule the nucleus possesses plenty of motion, only reduced 
to an eccentricity below unity. In the simile the "satellites" 
possess merely elliptic motion, whereas in the molecule they pos- 
sess hyperbolic motion. But still the simile may help the under- 
standing; and in it the juggler's energy of action must be sup- 
posed to be the energy inherent in the balls themselves. 

When the juggler exhibits little energy only a small portion 
of the balls will be kept in the air at once. The bulk of the stock 
of mass on hand remains on the table; that is, in the central 
nucleus of the molecule. As the juggler gains energy, however, 



154 ENERGY 

the flying balls will remain longer in the air, and the idle stock 
on hand must be drawn upon in order to supply fresh recruits. 
In this simile, the total energy of the flying balls (including their 
potential energy) is the heat of the molecule; their vertical 
kinetic energy per unit of mass is its temperature ; the distance 
they fly is its volume ; the force with which they might impinge 
upon a ceiling overhead is its pressure ; and the degree to which 
the total stock of balls is subdivided into possible pairs by the 
juggler's activity (calling the idle stock on the table a solidified 
unit) is its entropy. 

The following picture of the physical nature of entropy is 
taken from the author's article in the November (1908) issue of 
the Harvard Engineering Journal : 

"Every preparatory student understands entropy thoroughly, 
although he has never been taught to know it by name. For 
every school-boy knows well the value of organization for team- 
work upon the athletic field. He knows that, for good, effective 
sport, a given mass of men must be used, not as a single massive 
unit, nor yet as a number of independent individuals; but by 
subdivision, specialization and organization it must be animated 
into an organic whole. 

"Indeed, without subdivision into at least two sides there can 
be no game at all ; just as in mechanical energy there can be no 
energy until the mass present is subdivided into at least two 
portions, which mutually oppose and react upon each other. 
Similarly, the particular sort of energy called athletic antagonism 
consists in the reaction occurring hetzveen two opposed sides. It 
does not lie in either side alone, nor yet in the individuals alone — 
though each of the latter possesses his own fund of internal 
energy (due to his subdivision into a variety of organs, muscles, 
glands, etc.). For in a company of even the most stalwart 
athletes there could be embodied no other form of energy than 
muscular, glandular, etc., exhibited in individual feats, unless 
there existed athletic organization into at least two contending 
parties. 

"For in all the better sorts of athletic contests, such as base- 
ball and foot-ball, there arises a quite distinct form of energy 
from the organic energy of the individual. The Subdivision, 
specialization, organization and interaction between players 
applies not merely to the mass as a whole, but pervades each of 



TEMPERATURE AND ENTROPY 155 

the two 'sides.' One man trains to be a pitcher, another to be 
catcher, and a third to be short-stop. Each 'side' becomes itself 
an organism, containing a form of energy which is quite distinct 
from the aggregate energy of the individual players. The two 
sides cooperate competitively to the production of a game. The 
men on each side compete cooperatively to the advancement of 
that particular side of the game. There thus arises a more intri- 
cate and effective form of contest than is possibly to be had from 
any number of athletes whatever, trained to any degree of skill 
and strength whatever, if each acts only as an individual. There 
has arisen a new 'form' of energy : team-work. 

''This subdivision and specialization into team-work consti- 
tutes athletic entropy. Correspondingly, athletic temperature is 
the intensity and vigor of play of each individual player. Both 
are cultivated by the coaches, as essential to good results ; but 
they aid in those results in distinctly different ways. These ways 
might be contrasted, for instance — if hearsay be accepted as true, 
for the purpose — by saying that Harvard wins her victories by 
athletic temperature, by brilliant individual play, whereas Yale 
wins hers by athletic entropy, by a dogged devotion to team-work 
which overbears in the long run. 

"Or a similar contrast might be borrowed from the field of 
heat-engines. The gas-engine, with its tall and white-hot but 
slightly entropied cycle, represents the acme of efficiency in 
power-development from heat. Each molecule of its working- 
substance moves at tremendous radial speed, or temperature. 
But the gas-engine has never been able to compete success- 
fully, in the world's work, with the steam-engine; for the 
latter's squat, cool and slow-moving cycle is so heavily entropied 
that the engine stays in the ring and continues to push 
hard, long after the more delicate gas-engine has 'laid down on 
the job.' " 

This position is to be emphasized more as time passes. The 
importance of team-work is ever upon the increase. As popula- 
tions increase inventions multiply. A greater number of people 
are now in daily communication with each other — and therefore 
forced to cooperate, whether they will or no — than a century 
ago could be reached in a month's travel. Life, in shop, market 
and legislature, grows daily more multiplex and intricate. The 
need for the better and more patriotic understanding of human 



156 ENERGY 

relationships grows daily more urgent, while that for brilliant 
patriotism of individual deeds is on the wane. 

All this is inextricably tied up with this question of entropy, 
or mass-pairing, or the quantity-factor of energy. In such a 
book as this no more than a suggestion of its broad significance 
and importance can be made. But to pass the occasion without 
even that would be a grave mistake. Our best records in Ameri- 
can history are the victories won by national team-work. In 
1864 it was the solidarity of the North, in an organized effort of 
each for all, which won over the more intense valor and superior 
tactics of the South, which lacked it. In 1898 all the credit we 
won was due to national entropy, and all our shame to a lack 
of it — to the false idea that intense and spectacular Rough 
Riding could accomplish what was desired, without the coopera- 
tion of every individual in the nation to the army's proper feeding 
and nursing. 

The conclusions stated in the above pages are of fundamental 
importance. They may be summarized as follows : 

1. Temperature is the intensity-factor of heat. It is the 
radial kinetic intensity of the mass-particles, large and small, of 

velocity-squared 

the molecule. It is proportional to ——-, z z ^, or 

,_ total mass of molecule 

mathematically, to Il(-^). It is affected solely by changes of 

volume under pressure, contributing or abstracting radial motion 
only (or eccentricity of orbit) to the particles of the molecule. 

2. Entropy is the extensity-factor, or quantity-factor, of 
heat. It is the extent of mass-pairing, or degree of subdivision, 
of the molecule, into separate particles which interact ener- 
getically. It is the variable proportion of the mass of each mole- 
cule effective in heat-motion. Mathematically, it is S(MM). 

3. Labority is the isentropic alteration of temperature, by 
elastic, or radial, work-performance or absorption. It is a varia- 
tion of the intensity of thermal energy, or velocity of radial 
motion, under constancy of extensity, or degree of mass-pairing, 
or entropy. 

4. Thermogy is the isothermal alteration of entropy, by 
inelastic, or tangential, work or heat absorption. It is a variation of 
the extensity of thermal energy, or entropy, or subdivision of the 
molecule, under constancy of its radial intensity, or temperature. 



TEMPERATURE AND ENTROPY 157 

5. Both of these processes occur in apparent purity in nature. 
The first occurs in adiabatic expansion or compression — finding 
almost pure instance in the explosion of a steam-boiler, where 
lack of time excludes the conduction of heat to or from the 
outside. The second finds instance in the vaporization of water 
under constant pressure. 

6. All other thermodynamic processes than these consist of 
combinations of the two — either produced simultaneously, by two 
outside causes acting at once, or occurring seriatim, one as the 
result of the other, within the internal equilibrium of the mole- 
cule. Instances of the latter are the isomorphic heating of water, 
where the temperature rises because the entropy has been 
increased; or the wire-drawing of steam, where the entropy 
increases because the temperature has fallen. 

Instances of the simultaneous operation of two external 
influences occur in the compression or expansion of gases in 
actual cylinders. In compression the temperature is raised by 
work-performance and the entropy decreased by cooling, simulta- 
neously. In expansion the temperature is dropped by work- 
performance and the entropy decreased by cooling. Every pos- 
sible combination of thermogy, positive or negative, with labority, 
positive or negative, is known in power-house practise ; and none 
of them can be explained consistently without this clear distinc- 
tion between thermogy and labority, as independent, though some- 
times connected, mechanical phenomena. 

As possibly of further aid in understanding a difificult subject, 
the following parallel between temperature and entropy is given : 

Temperature. Entropy. 

In Molecular Mechanics: 

Intensity of heat-energy, or space- Extensity of heat-energy, or quan- 

and-motion factor. tity-factor of heat. 

Radial activity. Mass-pairing involved, maintained 

by tangential activity. 

Degree of radial space and motion Extent of subdivision of molecule 

between mass-particles of mole- into mass-particles radially sepa- 

cule. rated by space or motion. 
XJ2sin2 a c 

Proportional to e. j^ _^j^ =^-^- Proportional to 2 M1M2. 

In Relation to External Bodies: 
Action normal to body's surface. Action parallel to body's surface. 

Elastic work-performance or ab- Inelastic work-absorption, or ab- 
sorption, sorption or radiation of heat. 
Isentropic labority. Isothermal thermogy. 



CHAPTER XIII. 

The Energetic Cycle. 

Hitherto the discussion has turned chiefly upon what energy- 
is. It next becomes of importance to consider how it may be 
obtained for human consumption. The question is not one as to 
the sources from which energy may be derived. Much has been 
written upon this topic; yet, except for the need for endlessly 
repeating the lesson that the waves of the sea do not constitute 
a practicable source of supply, nor even the tides (except under 
most unusual circumstances), there is little to be said in this 
connection. 

The question of prime interest is not : How may man acquire 
energy ? It is, instead : How does nature do it ? 

On every hand, in every natural phenomenon, is seen an 
endless chain of energy-transformations. The one stock of 
energy possessed by nature is made available for the most intri- 
cate and endless variety of purposes and effects, merely by 
transformation. In the power-house one sees a little of this : 
The ownership of a store of black, dense, heavy, unchanging 
coal gives one permanence of potentiahty for power. Upon 
demand, this may be converted into heat and light, in combus- 
tion. But the permanence is gone; transferability, and also 
evanescence, have taken its place. 

The evanescence of flame-heat being usually too great, a fair 
degree of permanence is acquired, with portability retained, by 
transformation into steam-heat. This, upon demand, becomes 
mechanical motion. This, in turn, may become electricity. The 
electricity may then become chemical energy again, the form 
from which it started, in electrolysis ; or light and heat again, as 
it was in the furnace ; or an ether-wave carrying human intelli- 
gence across the midnight seas ; or mere motion and heat again. 
Even in the human laboratory the transformations are startling 
enough. But in Dame Nature's they are far more so ; they are 
stupendous, amazing and quite incomprehensible, except in their 
most general aspects. 

158 



THE ENERGETIC CYCLE 159 

To nature, therefore, the task of keeping all her vast and 
intricate processes in motion is not one of creation, or acquisi- 
tion, of energy, from some other source. It is merely that of 
keeping what she always possesses in active transformation and 
circulation. Does nature wish to grow a world of fresh green 
verdure, some spring, or rear an entire human race in some 
twinkling of the universal eye? She does not transport a 
vegetation or a mankind from some other corner of the universe. 
She merely places between the sun and that unknown space into 
which it has been radiating its heat, throughout unrecorded time, 
a planet: an earth, watered and aired and fitted for its mission. 
The result follows. Without necromancy, as naturally and mechan- 
ically as a properly made motor starts when the current is turned 
on, a flood of verdure spreads over the face of the earth when 
spring comes, or a fauna of pterodactyls and eohippi develops 
into races of horses or men when the changes of season become 
sufficiently favorable. 

It is quite similarly that man places his simple little con- 
trivances in the way of the same vast currents of energetic flow 
in nature, that he may consummate his own tiny and short- 
sighted plans. He finds a reservoir of water up amid the hills, 
and sees its contents cascade toward the sea. In between the 
lake and sea he places his water-wheels, and derives power. He 
finds a red-hot fire, radiating heat into the lower realms of 
temperature. In between fire and refrigerator he places his 
steam-engines, and derives power. 

The plan is a simple one to look at. But how does it work, in 
terms of the ideas as to heat and work which the preceding papers 
have outlined ? 

Take the simplest case first: that of the mill-pond, the tail- 
race and the old-fashioned overshot water-wheel between them. 

The water in the mill-pond possesses energy because of its 
separation from the earth. It and the earth constitute two mass- 
portions, which are paired or opposed in a mutual reaction 
which we call force, or "weight," and which embodies energy. 
The two were pulled apart originally by sun-heat, a "supply of 
energy from without," and they tend to reunite as they reship 
their fund of energy upon its next journey in the world. 

Man, however, cannot utilize an entire mill-pond of water at 
once. He therefore takes it a little at a time. A cubic foot of 



160 ENERGY 

water, or a ton, say, will be taken into the upper buckets of a 
water-wheel, to do work. This ton of water is detached from 
the natural supply, and introduced into the wheel; and in the 
performance of this process great care is taken to have no "free 
fall, friction or impact" occur, beyond what is unavoidable. 

This simple process of detaching energy-bearing mass from 
the stock of it which nature brings to our hands, and its embodi- 
ment into the machine with which it is designed to develop 
power, is simplicity itself. Yet it is necessary to call attention to 
it in this special way, because it typifies a process which is an 
essential part of all energetic cycles, and which appears to be, in 
thermodynamic cycles, one of the most difficult of all processes 
for the student to understand. 

When the stock of energy-bearing mass is once in the wheel, 
it is permitted to lose its vertical separation from the earth, by 
falling, under resistance and control, to the tail-race level. It is 
there detached from the wheel and rejected into the tail-race. 

According to the mathematics of engineering, if W be the 
weight of water taken into the wheel and h be the height of 
mill-pond above tail-race, the energy transformed by the fall is 
Wh. The cycle of events might be portrayed by a plain rec- 
tangular diagram, such as Fig. lO, in which the horizontal 
abscissae measure weight and the vertical ordinates measure 
height. If the point D should represent the weight and height 
of the empty buckets at the top of the wheel, DA the weight 
of the water taken in, OI^ the height of the mill-pond above the 
sea-level and OI2 the similar height of the tail-race, then the 
work done by the falling buckets would be measured by the 
rectangle I^ABIs and that by the rising buckets by the rectangle 
I^CDIg. The net difference, or the rectangle DABC, would 
measure the net work done, or Wh. The efficiency with which 

T)ABC T T 
the total available fall had been utilized would be -p^ . „ „ =777^- 

But under modern conditions the old-fashioned overshot or 
direct-gravity water-wheel has had to give place to the turbine. 
For handling large quantities of water under heads which often 
exceed the largest practicable diameter for wheels, the turbine 
is far superior to the older form. For very high heads and 
small quantities of water, such as occur in the mountains of the 
mining-regions, the Pelton wheel has been used in preference to 



THE ENERGETIC CYCLE 



161 



the submerged turbine. And as the different parts of the Pelton 
wheel are separated more distinctly to the eye, it may form a 
clearer illustration in the following discussion than the turbine. 

For in both of these modern types of water-motor another 
cycle of energy-transformations than that just described has been 
interposed between the mill-pond and the tail-race. The water 
performs its vertical fall, not in the moving machine, but in a 
closed penstock. At the foot of this penstock its energy has 
been stored in the form of accumulated pressure. It is then 
chiefly ''transient" energy, although a slight portion is stored' 
elastically in the compression which the water has suffered during 
its fall. 




OF ENERGY 



FIG. lO. 



Before the water is admitted to the moving part of the wheel, 
this transient and resilient energy is converted into kinetic energy 
of jet, by the nozzle or guide-blades. It is in this form that it 
is received by the wheel proper, which is so designed as to be 
adapted to the reduction of velocity-relatively-to-the-earth by an 
alteration of direction of velocity-relatively-to-wheel, by means 
of curved vanes. The illustration of the manner of doing this, 
by means of parallelograms of velocity, is familiar to every tech- 
nical student, and needs no reproduction here. 

The energetic cycle which is thus performed by the nozzle 
and wheel in cooperation is quite similar to that of the overshot- 
wheel, in principle, and may be illustrated equally well by Fig. 
lo. Only, in this new use of Fig. lo the vertical ordinates must 



162 ENERGY 

be accepted as measuring, not height nor head of static water- 
energy, but the velocity-squared of kinetic water-energy. The 
horizontal abscissae measure the quantity-factor, usually taken 
as mass, instead of weight, as before. 

In this new case it is as true as it was before that the energy 
taken into the wheel is measured by the rectangle DAX^Xg, that 
rejected to the tail-race is BCXgX^, and that converted into work 

DARC 
is DABC. The efficiency of conversion, as before, is .p^ . ^ ^ = 

Y^ . The only difference is that now there are no "empty 

buckets" to be carried up and down, and so the vertical axis OI 
should coincide with GD. 

In these two mechanical cycles, which are so familiar to all 
engineers that their description seems superfluous, there are 
visible all of the characteristics of the most obscure thermo- 
dynamic and other energetic cycles, if the positions taken in the 
preceding papers of this series be true. It is therefore important 
to observe carefully just what has happened therein. 

In the first place, the question must be reviewed: What is 
potential hydraulic energy? What is kinetic hydraulic energy? 

The equations which were given as exact answers to these 
questions, in Ghapter I, are these : 

Potential energy = c M1M2 (-^ q-) (32) 

Kinetic energy =^^1^2 ^^'^^^ (33) 

In these expressions, it was pointed out, the factor M^Ma 
indicated the extent to which the total mass present was sub- 
divided into mass-pairs capable of embodying energy, and was 
called the extensity of energy. The remainder of each expression 
indicated the degree of spacial or kinetic relationship which was 
embodied in these mass-pairs, to constitute them energetic pairs, 
and was called the intensity of energy. 

The study of the general characteristics of mass, space and 
motion, as they appear all about us in nature, showed that each 
of these factors might be an independent variable. The ex- 
tensity factor, for instance, varies in one direction by the con- 
solidation of mass, and in the other by its subdivision. The 



THE ENERGETIC CYCLE 163 

forces of gravitational attraction are everywhere and all the time 
tending to produce consolidation. The equally universal phe- 
nomena of motion, collision, impact and friction, engendering 
tangential motion and centrifugal force, are always tending to 
produce subdivision and comminution of matter. 

The intensity-factors also vary in either direction, and under 
the same forces or closely allied ones. Gravitational attraction is 
always tending to increase the propinquity between masses, and 
likewise their velocities. The centrifugal forces developed by 
those velocities are always tending to decrease the velocities and 
the propinquity simultaneously. 

Two Basic Processes of Mechanical Energetics. The 
two fundamental processes of mechanics, therefore, are: 

1. The variation of mass- pairing, or extensity of energy, by 
the subdivision or consolidation of mass, under constancy of 
intensity ; 

2. The variation of velocity or propinquity, or intensity of 
energy, by the approach or separation, or by the acceleration or 
retardation of masses, under constancy of mass-pairing. 

The second of these processes is apparently the more familiar. 
The performance of work by falling weights, or by hammers 
"slowing up" against anvils or by trains climbing grades by 
momentum, or by water accelerated in nozzles and retarded 
against moving vanes, are all familiar instances of variations in 
intensity of energy under constancy of mass-pairing. 

But mixed in with and confused with these processes is the 
first named sort, which are distinctly contrasted with them in 
character. That is to say, before the weight can fall and perform 
work it must first be released, or "detached," from the earth, so 
as to form a mass-pair reacting with it. The hammer, before it 
can be swung on the anvil, must first be picked up. The water, 
before it can project against vane, must first be detached from 
earth and mill-pond. The railroad-train, before it can climb a 
grade by its own motion, must first be built and constituted a 
thing separate from the earth : ore must be mined and smelted, 
designs made, metal cast, wrought and machined, and the parts 
assembled into a complex whole. The complete structure must 
be equipped with water, fire, oil, etc. All this must precede the 
purely mechanical process of setting the train into motion fit to 
climb a grade. 



164 ENERGY 

In these examples it is clear that great differences in appear- 
ance occur in this one process of subdivision of the originally 
unit earth into a mass-pair capable of embodying energy. In the 
tripping of the drop or the picking up of the hammer it is so 
slight and incidental a thing that it is commonly overlooked as 
being a mechanical process at all. In the case of the manufacture 
of the railroad-train it is so complex and important that a whole 
string of factories is required for its performance. 

The same thing is true of the different sorts of energetic 
cycles. The processes DA and BC are variations in the ex- 
tensity of energy : The detachment of the working water from 
unity with the earth, in the first place, and its reconsolidation 
with the earth in the second. The processes AB and CD are 
variations in the intensity of the mass-pair, after its formation 
or destruction by the variations in extensity. In the hydraulic 
cycles these variations in extensity are commonly overlooked, as 
essential parts of the cycle; and yet they are difficult enough to 
perform properly. The water must be gotten into the overshot 
wheel with as little free fall, impact and friction as possible, and 
gotten out again in the same way. In the turbine or Pelton 
wheel the supply of motion-bearing water must be transferred to 
the vanes with as little impact and friction as possible, and the 
water must leave the vanes for the tail-race in the same way. 
The skill with which these extensity-varying processes are car- 
ried out often has as much to do with the efficiency of the 
wheel as has the skill with which the intensity-varying processes, 
by which the height or velocity of the water is reduced within the 
wheel, are performed — and in the former case usually much more. 

It is in this careful way that the most familiar mechanical 
processes of the shop and the power-house must be analysed and 
understood, if they are to form a means for understanding the 
more obscure cycles of molecular mechanics. For, as the different 
forms of energy and methods of cyclical action are taken up, one 
after another, hardly any two will be similar in their external 
appearance. One will emphasize the DA-process, or the prepara- 
tion of the mass-pair ; as in the building of the railway-train and 
road. Another will emphasize the AB-process of dropping the 
intensity under control, as in the turbine water-wheel. Another, 
like the overshot water-wheel, will be excluded from practica- 
bility by Its exaggeration of the CD, or empty-bucket, work, in 



THE ENERGETIC CYCLE 165 

proportion to its net work DABC ; while in the Pelton wheel the 
CD-work will be entirely absent. Yet the underlying principles 
are in every case the same. And as the study of thermodynamic 
cycles is reached this dissimilarity of appearance and identity of 
principle becomes most marked. 

The first step toward these more obscure cycles is to note that 
the mathematical statements concerning the static and kinetic 
hydraulic cycles already given are not quite correct. That for 
the overshot water-wheel assumed that the weight of the water 
would be constant for all heights above the earth, which is not 
quite true. That for the Pelton wheel assumed that the kinetic 
energy was proportional to the square of the velocity, whereas 
the sum of the masses present must be inserted as a divisor. The 
trtle expressions for the intensity itself appeared in Equations 24 
and 25. 

In order, then, to have Fig. 10 become exactly and generally 
true, its axes must be understood as measuring, not weight and 
height, or mass and velocity-squared, respectively, but the true 
factors of intensity and extensity of energy, respectively. The 
ordinates must measure intensity, the abscissae extensity, of 
whatever form of energy is considered. That the mind may 
effect this transfer of ideas from the similes already employed, 
four mental substitutions from the approximate to the exact 
must be made. These are: 

1. The substitution of the exact extensity-f actor, or degree 
of mass-pairing or mass-product, for its approximate manifesta- 
tion, weight. 

2. The substitution of the exact spacial intensity-factor, or 

propinquity = c '— , for its approximate manifestation, height. 

3. The substitution of the exact kinetic intensity-factor, or 
velocity-squared-divided-by-mass, for its approximation, velocity- 
squared. 

4. For heat-energy, the concept of Fig. 10 as portraying 
both of these intensity-factors simultaneously, for any given 
complex mass-system, instead of merely one alone. 

This careful line of attack upon the mysteries of the thermo- 
dynamic cycles has been long and perhaps tedious ; but the fruit 
is worth the trouble. The typical, or pure, thermodynamic cycle, 
the Carnot cycle, now appears as nothing more mysterious than 



166 ENERGY 

the cycles of the old-fashioned overshot and the newer Pelton 
water-wheel, in intricate combination; somewhat disguised by 
the number, minuteness and intricacy of the motion-bearing mass- 
pairs which supply the energy, it is true, but in no wise more 
mysterious in its principle of operation. 

In 1824 Sadi Carnot, in his "Reflections upon the Motive 
Power of Heat," laid dow^n these rules for the performance of a 
thermodynamic cycle with the maximum possible efliciency be- 
tween any given temperature-limits :* 

1. The heat must be taken in isothermally, at the tem- 
perature of supply ; 

2. The heat must then drop its temperature, performing 
work, isentropically ; 

3. The heat then rejected must pass to the refrigerator, or 
absorbent, isothermally, at the temperature of the refrigerator ; 

4. The remnant of heat still left must be raised to the tem- 
perature of the original supply isentropically, absorbing work. 

The four processes thus defined are seen to constitute two 
pairs. Each pair consists of examples of one of the two thermo- 
dynamic processes which were defined, in Chapter XI, as the 
only basic ones. One of these was the isothermal variation of 
entropy. The other was the isentropic variation of temperature. 
And these two fundamental thermal processes are identical with 
the two fundamental mechanical processes zvhich were defined 
and numbered on page 163. 

That is to say, the four processes of the Carnot cycle may be 
represented by Fig. 10 with perfect accuracy, if the two axes be 
considered as measuring absolute temperature in the vertical case 
and entropy in the horizontal. Fig. 10 represented the kinetic 
mechanical cycle with equal accuracy when these two axes were 
considered as measuring intensity of space and motion in the 
vertical case, and extensity of subdivision of matter in the hori- 
zontal case. The only things needful to be kept in mind, in order 
to identify the two cycles completely, are three in number. In 
the first place, the distinction between radial and tangential 
mechanical action within the molecule must be understood. 



*The language has been modernized into accord with the terms already 
made familiar to the reader in the preceding pages. Carnot's language was 
no less clear and explicit, except that he confused the ideas of 2 M and 
2 MM — a confusion which is still widespread among teachers of energetics 
today. 



THE ENERGETIC CYCLE 167 

Work can be performed only by radial mechanical or molecular 
energy. 

In the second place, comes the simple, but confusing, question 
of locality in which the several processes are performed. 

In the third place, any thermodynamic cycle, such as the 
Carnot, comprises both forms of mechanical cycle : the kinetic 
and the static, in its molecular action — just as a cannon-ball at 
the top of its trajectory will do work in virtue of both its height 
above the earth and its horizontal velocity too. 

As to locality, in the hydraulic cycle, for instance, the cycle 
naturally starts with the process DA: the acquisition of energy- 
bearing mass. No thought is taken as to how nature may have 
prepared this natural supply of energy-bearing mass. Those 
processes lie far outside our water-wheels, and we overlook 
them. In the thermodynamic cycles, on the other hand, the 
preparation of the energy-bearing mass occurs within our power- 
houses, although not within the engines. The preparation of the 
steam in the boiler-room must be cared for and understood, as 
well as its work-performance in the engine-room. In the boiler- 
room the heat conducted through the boiler-shell acts tangentially 
upon the water-molecuk's particles. At first, as the water is 
heating, the breaking up of the molecule by this action is quickly 
annulled by its compression into liquidity, by the superior pres- 
sure acting on the surface of the water. As each molecule 
attains sufficient internal whirling velocity, however, so that its 
centrifugal forces are able to overcome these external forces, 
the water-molecule pops into a steam-molecule. In order to give 
it its separateness, while at the same time conserving its same 
centrifugal force of whirling, or its "pressure," as we should say, 
with enormous increase in volume — a task similar to elevating a 
cannon-ball to the highest point of its trajectory while conserving 
its original muzzle-velocity — a large quantity of energy is 
absorbed, in latent form. The steam-molecule, thus charged with 
kinetic energy in the form of sensible heat, or temperature, and 
wath space-energy in the form of volume and latent heat of 
vaporization, is then carried over to the engine, where it gives 
up both forms of energy, in part, to the piston. 

The piston, it is true, is adapted only for the absorption of 
the kinetic energy, or temperature-heat ; but while the kinetic 
intensity of the steam-heat is no greater than that of water of 



168 ENERGY 

equal temperature, the extent of kinetic energy, or the number 
and mass of the projectiles which are in a condition to impinge 
upon the piston has been vastly increased by the vaporization. 
A water-molecule engaged in work-performance is like a battle- 
ship firing one gun at a time. The steam-molecule is like the 
same ship firing a broadside. The intensity of projectile-energ}' 
(not the "intensity of fire," as artillerists use the term) is the 
same for one gun as for twenty of the same calibre. For the 
broadside the muzzle-velocity is no greater, but the effectiveness 
of fire — the "heft" of the blow against the enemy — is far greater. 
The oft-heard but mistaken argument that steam-heat is 
inefficient for work-performance because of the latent heat in 
the exhaust is exactly parallel to the possible objection to broad- 
sides as a waste of ammunition. If the enemy be present in 
form fit to absorb a broadside, it is vastly more efficient to 
employ broadsides than a dribble of single fire. The work is 
done promptly, powerfully and efficiently. And in building our 
steam-engines there is no difficulty in making the pistons fit to 
stand the broadsides. The powerful steady pressure of the latent 
heat of the steam is a vastly more useful laborer, taking the run 
of all conditions, than is the highly intense, but evanescent, pres- 
sure of the gas-engine cycle. The mere fact that latent heat 
costs something, as do broadsides, has little to do with the case. 
Many of our large gas-engine builders are to-day modifying their 
gas-producer accessories in their power-houses so that their 
engines may have more entropy to work with ; though to tell 
them that this was what they are unconsciously doing, or that 
entropy were worthy of any engineer's serious study, would 
probably fill them with astonishment and disdain.* 



*Not only is it not true that latent heat is inimical to efficiency of work- 
performance, but it is true that no other form of heat is equally efficient. 
Thus, using water and steam as the illustrative working-substance, Fig. B 
shows the Rankine cycle of the ordinary steam-engine, using saturated 
steam, including the heating of the feed-water in the boiler. The 
heating of the feed-water is shown by the isomorphic curve AB, the 
vaporization of the hot water by the isothermal BC, the work-performance 
in the engine-cylinder by the isentropic CD, and the condensation in 
the condenser by the isothermal DA. The total area ABCDA measures 
the heat converted into work. Of this, that derived from the sensible 
heat of the water is the triansrular area ABE ; that derived from 
the latent heat of the steam is the rectangle BCDE. From this it is 
obvious that, for equal ranges of temperature and entropy, the efficiency 
of sensible or isomorphic heat for work-performance is less than half that 
of latent heat 



THE ENERGETIC CYCLE 



169 



As the molecule expands against the piston, doing work, it 
loses kinetic energy under constancy of mass-pairing; that is to 
say, the fastest moving particles are slowed down, by rebound 

Indeed, could we build practicably an engine which would follow the 
cycle BCDE, using only latent heat, we should have it operating upon the 
Garnet cycle, the one cycle of maximum efficiency. 




FIG. B. 

Again, compare with the above the Beau de Rochas cycle of the ordi- 
nary gas-engine, as shown in Fig. C. Herein the total temperature-range, 
from A to C, is usually some 3600° F. Since the absolute temperature at 
A is about 600", the best efficiency obtainable from the temperature-range 
available would be 3600-^4200=0.86. The efficiency of the Rankine cycle, 




FIG. C. . 

under common ranges of pressure and temperature in condensing engines, 
or from about 600° to 800° absolute, would be similarly computed as nearly 
0.25, Of this 25% available the steam-engine actually develops some 
three-fifths, or 15%. But of the 86% available for the gas-engine the latter 
seldom exceeds about three-tenths, or 26%. The difference is due entirely 
to the poor form of the gas-engine cycle. The trouble is that its heat is 
all isomorphic or sensible in form. In other words, were it practicable 
to build a heat-engine having the good points of the gas-engine, but using 
latent heat instead of isomorphic heat, its efficiency would be about twice 
that of the best standard gas-engines. 



170 ENERGY 

from a retreating piston, but without any direct tendency to 
diminish their number. That could be done, as was shown from 
the elementary mass-pair, only by a tangential retardation at 
apastron (the point of ''contact" with the piston). The loss of 
radial component against the piston — which component is the 
only one which motion of the piston might affect, and which 
alone constitutes temperature — tends only to decrease the eccen- 
tricity of orbit and increase the periastron distance. The satellitic 
particles change their orbits from cometary forms, which whirl 
closely about the central nucleus and then shoot into space with 
great and almost radial velocity, to orbits like those of the outer 
planets : slow, remote and almost circular. The change is like 
that from a child's return-ball, shooting out and back on an 
elastic thread, to the same ball whirled about the hand on a 
string of fairly fixed length. And the "hot body," as the swarm 
of such molecules is called, becomes expanded, cooled, rarefied 
and of decreased expansive (or radial) pressure. 

It happens, in the case of steam, that as this process goes on 
the form of molecular structure leads to a lack of internal 
equilibrium, so that, in order that the majority of the molecules 
may maintain the form described, a few are forced to recondense 
into water — partial condensation being a well-known accompani- 
ment of the adiabatic expansion of steam. But this fact is 
merely an accidental characteristic of water. With ether, for 
instance, the exact opposite is true. Adiabatic expansion leads 
to superheat. Such facts as these, therefore, have no bearing 
upon the general idea that work-performance by heat is nothing 
more than a transfer of purely mechanical energy from the 
m.olecules to the piston. 

For instance, in the nozzle of a steam-turbine of the de Laval 
type — and the nozzle is the only portion of this steam-turbine 
where thermodynamic action takes place — this mechanical 
explanation of adiabatic expansion is even more obvious. Here, 
instead of a retreating piston being opposed to the radially flying 
particles, nothing is opposed. The nozzle is merely a device for 
removing from the steam all resistant pressure in one particular 
direction, the forward one. The flying particles which happen to 
depart from their molecular nuclei in other directions than this 
meet the solid walls of the nozzle, and return with orbits unal- 
tered. But those which happen to fly in the direction of the axis 



THE ENERGETIC CYCLE 171 

of the nozzle find nothing to return them ; and as they all possess 
hyperbolic orbits, they depart from home forever. Instead of 
giving up their radial velocity to a piston, they keep it. Only 
now it has become a linear velocity, not only in reference to the 
nucleus but also in reference to the walls of the nozzle. The 
swarm of molecules strings out into a long procession, like a 
body of troops defiling into column-formation; and although the 
cross-section has diminished, the volume has increased. The 
molecules find more '*elbow-room." And while their original 
radial velocity remains unchanged, relatively to each other it has 
disappeared. Only the tangential component remains, giving to 
the orbits the same wide circularity which they had after colli- 
sion with the retreating piston. 

The impinging of this jet of molecules upon the curved vanes 
of the turbine-wheel, and the performance of work there by the 
loss of linear velocity, is a purely mechanical action, alike in 
principle in hydraulic and steam turbines. The expanded mole- 
cules pouring forth from the nozzle against the vane might just 
as well be so many microscopic but inflated foot-balls — originally 
fed into the boiler as so many solid wads of leather, and there 
inflated with compressed air by the energy of the furnace, and 
which have now escaped with a vigor of motion derived from 
their own internally stored energy — as expanded steam-molecules, 
so far as any thermodynamic action upon the vane is concerned. 
It is true that in actual practice thermodynamic action nearly 
always trespasses beyond the nozzle and modifies the vane-action 
in ways which cannot occur in hydraulic turbines. But this is a 
thing which the steam-turbine designer usually seeks to avoid, or 
to reduce to a minimum. 

Cycle-efficiency. In any rectangular cycle, such as Fig. lO, 
the energy taken in is proportional to the initial intensity, I^, 
whatever the width of the cycle horizontally. Usually, in nature, 
cycles are truly rectangular only when considered of infinitesimal 
width, dX. But since any cyclical area, however irregular in 
outline, can be divided into an infinite number of vertical strips 
of width dX between parallel sides, there is always a definite 
initial and final intensity of rectangular cyclical action, for each 
increment or decrement of energy handled. 

The energy rejected by the cvcle is similarly proportional to 
the intensity of rejection, lo. Therefore the efficiency of a rec- 



172 ENERGY 

tangular, or perfect, cycle, or the proportion of the energy taken 
in which undergoes transformation, is given by the expression 

F=iy^ (34) 

which is the fundamental equation for all pure energy-trans- 
formation. 

In mechanical C3^cles these initial and final intensities are pro- 
pinquities, or velocity-squared-divided-by-mass, as already defined. 
In thermodynamic cycles they are temperatures. In electro- 
mechanical cycles they are voltages. But whatever the form of 
energy may be the law stated in Equation 34 holds true. 

Since the initial intensity, in natural action, can never be 
infinite, nor the final intensity zero, no cyclical action can ever 
convert all of any primary fund of energy into a secondary form. 
It will be shown later that energy-transformations are known 
between almost every two of the many known forms of energy; 
but in none of these many or diverse cases can the transformation 
be complete. Although it is commonly said that, whereas heat 
cannot be changed completely into work, yet work can be changed 
completely into heat, nevertheless this statement is not true, as 
will be shown later — except in a special, local and approximate 
sense which, while useful enough for certain purposes, has no 
place in generalities concerning energetic action. Indeed, we can 
acquire no proper understanding of energetic action until we 
understand that, not only is all of any energy-form never trans- 
formed into another form, but the difficulty of further continuing 
the transformation approaches infinity as the energy still awaiting 
transformation approaches zero. 

Reversibility. Upon the surface of the earth, when dealing 

with solid or viscous bodies, only the purely vertical of these 
energetic processes are obviously "reversible." Therefore the 
rectangular cycle, of maximum efficiency, has usually been defined 
in terms of "reversibility." But in truth, speaking more gen- 
erally of the principles of energetic action, not only do the 
vertical processes never occur in purity, or more than temporarily 
reversed, but the horizontal ones are equally reversible — as will 
be brought out in Chapter XV. As a temporary aid to the 
student energetic reversibility is a useful idea ; but as a cosmic 
principle it must be handled with great care. 



THE ENERGETIC CYCLE 173 

Such, in general, is the pure energetic cycle. It consists of 
four processes, viz : 

I. The subdivision of mass, at a high degree of intensity. 
This subdivision may occur bodily, as in splitting off from the 
mill-pond the energy-bearing water which enters the water- 
wheel; or it may occur indirectly, as when a molecule of water 
within a steam-boiler is subdivided by tangential contact with the 
rapidly moving molecules of the hotter boiler-shell. In either 
case the work-performer (the water-wheel or steam-engine) 
receives mass which may be called ''energy-bearing" because it 
is in a state of subdivision, with an intensity of either separation 
or of motion (or both, as in the case of steam) between every 
two particles formed by this subdivision. 

II. The loss of intensity under constancy of subdivision. 
In water-wheel or engine-cylinder or steam-turbine nozzle alike, 
the number and size of mass-pairs at work is not affected by the 
performance of work. It is merely their intensity of relationship 
which is affected. 

III. The consolidation of mass at a lower degree of intensity, 
the lowest which it is practicable to attain by work-performance. 
In the water-wheel this consists in dumping the now useless 
water into the tail-race. In the steam-engine it is the condensa- 
tion of the steam in the condenser. 

IV. The raising of intensity of the now partially consolidated 
mass, under constancy of subdivision, to the original intensity — 
which, because of the consolidation, takes less energy than was 
given out in the third process. This is the most obscure of all of 
the processes, because man usually leaves nature to do it for him, 
and she works in obscure and intricate ways. 

The performance of these four essential processes requires 
the presence of four essential pieces of apparatus, viz : 

I. A supply of energy-bearing mass of high intensity of 
energy: the mill-pond for the water-wheel, the steam-boiler and 
furnace for the steam-engine, the chemically stored energy in the 
explosive gases of a gas-engine charge. In the first case the 
mass is transferred bodily from source to machine ; in the second 
case it is merely the subdivision which is transferred, by "con- 
tact." In the third case occurs no transfer at all, of either matter 
or entropy. The chemical subdivision between carbon, hydrogen 



174 ENERGY 

and oxygen is transformed directly into thermal subdivision, 
within the molecule, upon ignition. 

II. A device for dropping this intensity under control: in 
an overshot wheel the vertical motion of the buckets; in a 
turbine the curvature and motion of the vanes; in a steam- 
engine the retreat of the piston under pressure; in the steam- 
turbine nozzle the open end and conical walls, with the curved 
vanes beyond. 

III. A device for absorbing the waste energy of lowered 
intensity: in the water-wheel the tail-race; in the steam-engine 
the condenser, with its stream of cold water. 

IV. A means for the return of the rejected and consolidated 
mass to its original condition: in the water-wheel the sunheat, 
clouds and rain ; in the steam-engine the feed-pump and furnace- 
heat. It is this process which man, in all his prime-movers, 
leaves to nature to perform for him. Even in the case of the 
steam-boiler it is nature who is relied upon to furnish the coal 
to keep it hot. 



CHAPTER XIV. 

Reversed and Irregular Cycles. 

The cycle which was described in the preceding paper, 
whether relying upon a water-wheel or a steam-engine for 
illustration, was considered always as passing through the four 
processes in an order which followed a clockwise direction of 
passage around the diagram. Such an order of procedure leads 
always to the drawing upon nature for a supply of energy of 
high intensity of one form, to the rejection of a remnant of 
that energy (with all the mass, or mass-pairing, which carried 
it) at a lower intensit}^, and the supply to the operator of work 
at a usefully high intensity. 

It is obvious, therefore, that cyclical action is devoted to the 
transformation of energy, from one form which nature supplies 
to another which man desires. The first of these forms will be 
called the primary form and the other the secondary form. In 
the case of the overshot water-wheel the primary form is static 
gravitational energy, or space-energy, between mill-pond water 
and earth, while the secondary form is the transient energy of 
the wheel-shaft, which may be converted into any of many 
forms within the mill. In the hydraulic turbine-nozzle the 
primary form is static space-energy, as before, but the secondary 
form is kinetic water-earth energy of jet. In the vanes them- 
selves of the hydraulic turbine the primary form is the kinetic 
water-earth energy received from the jet, and the secondary 
form is the transient energy of the wheel-shaft, as before. In 
the nozzle of the steam-turbine the primary form of energy is 
steam-heat of high temperature, involving both space and motion- 
energy between the particles of the molecules, as already 
described, while the secondary form is kinetic steam-earth energy 
of jet. In the steam-turbine vanes the primary form is kinetic 
mechanical energy of flying steam, while the secondary form is 
the transient energy of the turbine-shaft. 

A moment's consideration will make it plain that there is a 
very close relationship between the individual links of any of 

175 



176 ENERGY 

these chains of cycles. They are indissolubly linked. Thus, 
the water-wheel cycle cannot be performed unless there be a 
device for absorbing its power. Ordinarily this device was a 
pair of mill-stones, which absorbed it in friction. As this is too 
indefinite for our purpose, let it be supposed that the water- 
wheel drives a pump which draws water from the tail-race and 
discharges it into the mill-pond. Then, if the picture is to be 
complete, no water-wheel cycle could be drawn without drawing 
also a pump-cycle. And the pump-cycle, if friction is to be 
neglected for the moment, would be of equal area, or power, 
with the driving cycle. 

But the pump-cycle must be portrayed in a reversed, or 
counterclockwise direction. It receives mass-pairing at a low 
intensity of energy, raises its intensity, discharges it into the 
pond (or consolidates it with the earth), and then goes back for 
more. Such a cycle would be shown by Fig. lo, if the diagram 
be traversed in the direction CBAD. The reversed cycle, there- 
fore, is one in which the primary energy is received at low 
intensity and the secondary form discharged at high intensity. 

It is merely a corollary of the law of the conservation of 
energy to state that for every clockwise or direct cycle which 
takes place, there must be performed a counterclockwise or 
reversed cycle of equal area, to absorb its energy. No water- 
wheel can operate without a pump or equivalent to absorb its 
power. 

In portraying the action of any energetic machine, therefore, 
it is merely a matter of choice whether we portray the direct 
or the reversed cycle which occurs there. In the Pelton-wheel 
penstock and nozzle, for instance, there occurs a direct cycle of 
static hydraulic energy, which enters at high head and leaves at 
low ; but there is simultaneously performed a reversed cycle of 
kinetic hydraulic energy, for the water enters at a low velocity 
and leaves at much higher speed. The cycles are like coiled 
springs. The unwinding of the gravitational cycle winds up the 
accelerative cycle; and the unwinding of the accelerative cycle, 
within the wheel, a moment after, winds up some equivalent 
cycle of a further form. 

Thus nature works in an endless chain of transformations, 
by cyclical action, one thing giving up its strength that the next 
may live, and the next called upon a moment later to undergo 



REVERSED AND IRREGULAR CYCLES 177 

the same process of reproduction. In the power-house is seen 
quite a chain of such cychcal actions. The energy enters in the 
form of high-intensity chemical energy, in the coal-pile and 
atmosphere. This cycle unwinds in combustion, winding up 
simultaneously the cycle of thermal intensity, in high-temperature 
heat. This, in reality, unwinds again as its energy is radiated 
and conducted through the boiler-shell, though the fact cannot 
be seen unless it is analysed mechanically ; and simultaneously 
it winds up a similar cycle, though' a slower and more massive 
one, in the heat of the steam. This cycle unwinds again in the 
engine-cylinder, winding up the mechanical cycle of the engine- 
shaft. Or, if a steam-turbine be used, there are two transforma- 
tions within the engine, one in the nozzle or guide-blades and the 
other in the vanes of the rotor. 

So the energy might be followed upon its way, through elec- 
trical and other forms ; and everywhere, so far as can be seen, 
the chain of unwinding and winding up continues endlessly. 

In human affairs man is usually more interested in the 
unwinding, or direct, cycles; for he needs power more than he 
needs absorbents of power. But to a wide extent he also uses 
the latter. Pumps, elevators and refrigerating-machines all 
belong to this class. The cycle of the pump and the elevator is 
as simple, though reversed, as that of the overshot water-wheel, 
and needs no explanation. The cycle of the refrigerating- 
machine is much more obscure, however. For its details the 
reader must study the refrigerating-machines in detail. All that 
can be said here is that such a machine pumps low-temperature 
entropy (not heat) from the cold-storage room, or the water to be 
frozen, and discharges it at high temperature into any convenient 
waste, exactly as a pump picks up low-level water and discharges 
it at a higher level. The analogy is scientific and exact. It has 
already opened to our understanding wide fields of most useful 
progress in the arts, which await only a slight advance in our 
economic organization to make thoroughly practicable. We refer 
to the supply of heat for buildings upon a large scale by literally 
pumping up-temperature the low-temperature entropy which 
surrounds us during the winter months — perfectly good entropy, 
all of it; only a little too low on the temperature-scale for our 
use. Now we rely upon heat to heat our homes. Soon we shall 
reply upon power for that purpose ; not by converting it into 



178 ENERGY 

heat, in friction, but in pumping out-door heat far enough up- 
temperature to be agreeable to human nerves.* 

Irregular Cycles. So soon as attempt is made to carry out 
in practice any of the cycles described in connection with Fig. lo, 
it appears that it is impossible to do so, in purity. Nature never 
performs either of the two processes which were described as 
the basic energetic ones, the horizontal and vertical ones respect- 
ively, in purity. Each is always adulterated by the presence of 
the other to some degree. 

To explain, let Fig. ii represent an energetic field in which 
the sources and absorbents of energy are situated at the levels 
Ii and I2, respectively. These may be imagined as mill-pond and 
tail-race, if desired ; but the diagram applies equally to any of 
the more obscure forms of energy. If, in such a field, it is 
desired to operate a direct cycle, such as DABC, it proves to be 
impossible, with exactness. In order to effect the transfer from 
the sources of supply to the machine, the energy must be taken 
in, in whole or in part, at levels somewhat below DA, as along 
da. Similarly, it must be discharged at levels somewhat higher 
than BC, as along he. The area of the cycle dahc is therefore 
less than that of the cycle DABC, the cycle described by Carnot 
as the one of maximum efficiency. 

Similarly, the attempt to carry out a reversed cycle, such as 
CBAD, between these intensity-levels develops the fact that the 
energy must be taken in at some level below CB, as along CGB, 
and discharged at some higher level than AD, as along AHD. 
This cycle therefore absorbs more power, as measured by its 
area CGB AHD, than is stored by it usefully, as measured by the 
area CBAD. Its efficiency, too, is below the maximum possible 
with rectangular cycles. j 

The irregular direct cycle dahc is like that of a water- 
wheel into which the water leaks while the buckets are still 
rising or after they have begun their fall, or out of which the 
water leaks before the fall is completed or after return has 
begun. The irregular reversed cycle CGBAHD is like a pump 



*Ten a man who is spreading- his hands before a blazing fire on a win- 
ter's night that what he is absorbing and enjoying is not temperature, but 
entropy, and the speaker would probably sufifer. Nevertheless, it is true. 
Man takes almost enoueh bodily comfort, as well as industrial profit, out 
of this much abused, ignored and contemned drudge, entropy, to justify 
the proverbial saying that "ignorance is bliss." 



REVERSED AND IRREGULAR CYCLES 



179 



which draws in at a level below that of supply, and discharges 
above the waste-level. It is like that of a hod-carrier who should 
be set to carrying bricks from the street-level to the third floor, 
but who took his hod into the cellar to drop the bricks into it, 
then carried it to the fourth floor, and then dumped the bricks 
back to the third floor. 

From these considerations has been enunciated the general 
principle of energetics that the rectangular cycle of pure pro- 
cesses (as defined elsewhere in these papers) is the cycle of 
maximum efficiency. Carnot was the first to define this law, 
and he spoke in terms of thermodynamic cycles only ; but the 
material for its application to mechanical cycles has existed ever 
since the work of Newton, and that for electrical applications 
ever since that of Faraday and Ohm. 




FIG. II. 



If a direct cycle working between the limits I^ and 1^ should 
be set to operate a reversed cycle (as always occurs in nature), 
it is plain that the direct cycle must take some irregular form, 
such as dabc, which is inscribed within the rectangle DABC. 
The reversed cycle which is operated from it must in turn be 
inscribed within the rectangle DABC. Therefore the limits of 
intensity between which the reversed cycle is finally eflPective 
must be continually lower and lower ones, such as I3 and I4, 
which lie between I^ and Ig. It is this fact which has led to the 



180 ENERGY 

broad doctrine that the availability or intensity of the world's 
stock of energy is steadily declining. 

Nevertheless, the statement is not true. It was based upon 
too obscure a form of cycle, the thermodynamic, for all its 
bearings to be clearly seen. So soon as it is referred to a form 
where all the details can be followed, as in mechanical energy, 
its falseness appears as unquestionable. For then the "degrada- 
tion of availability" turns out to be based solely upon reference 
to a single form of energy — upon a form dictated solely by 
temporary human desire — and not upon any broad natural 
principle. Such a foundation is entirely too narrow to support 
a fundamental cosmic law. For what human beings preemi- 
nently desire seems to be rectilinear, or radial, motion. They 
have little use for tangential motion, except as it can be con- 
verted into rectilinear. 

Thus, observe a steam-boat ploughing through the water. 
In the engines, in the steam within them, in the boat itself, 
and in the water surrounding the boat, is an intricate mixture 
of radial and tangential motions. Ultimately it all becomes vir- 
tually tangential motion within the water^ in tiny eddies which 
constitute hydraulic resistance. These, later on, become still 
tinier molecular eddies called low-temperature heat. But one 
stage in the progress of the energy to this destination consists 
in a rectilinear, forward motion of the ship. Of all the motions 
present, this alone man esteems. He uses it as his measure of 
"efficiency." But nature esteems all motions equally, for she 
is able to reconvert the tiny eddies into rectilinear motion when 
she wishes to do so, as man cannot. 

In the economy of nature the readjustment of equilibrium is 
being constantly made by means of these chains of direct and 
reversed cycles of energetic action. The currents of energy are 
fairly continuous, seldom starting or stopping abruptly. The 
portions of matter through which they must find their way, 
however, are limited in their mass and dimensions. Therefore 
each portion of mass, in order to perform its allotted task of 
energy-transformation, must act over and over again. This it 
does by the method of the cycle. 

The ubiquity of these chains of cycles is too great for com- 
prehension. Wherever energetic action occurs, there proceed 
these chains of cycles. We have twice referred to that visible 



REVERSED AND IRREGULAR CYCLES 181 

in the series of machines and processes incidental to the modem 
power-house. But is it reahzed that each tiny molecule, not to 
mention smaller portions of matter, in all this apparatus is itself 
constantly carrying on its own particular chain of cycles, peculiar 
to itself? It is the integration of these countless hordes of less 
than microscopic cycles, into something big enough for man to 
see, which constitutes that major chain of phenomena which 
characterizes modern power development and distribution. 

But the power-house and its accessories are but a tiny instance 
of nature's broad dependence upon cyclical action. For a single 
instance, observe the interaction between vegetable and animal 
life. All vegetation is continuously operating a thermochemical 
cycle in a single direction. Drawing high-intensity radiant 
energy from the sun and low-intensity chemical energy from soil 
and air, in the form of the very stable chemical compounds, 
water and carbon-dioxid, respectively, it operates a direct thermal 
cycle to keep in operation a reversed chemical cycle. It main- 
tains a low normal temperature automatically, so that we seek 
the "cool green shade" on summer-days; and it stores up chem- 
icals of a higher intensity, in the form of starch, sugar and 
similar nutrients. 

In apposition with these cycles, all animal life maintains their 
obverse. Animals absorb the starch and sugar and operate a 
direct chemical cycle in their degradation into carbon dioxid and 
moisture. They do this in order to keep in operation reversed 
mechanical and thermal cycles, resulting in animal motion and 
high-temperature animal heat. 

Thus each half of animate existence here on earth both sup- 
ports, and at the same time properly loads, controls and balances, 
the other. Without the cooperation of the other, neither could 
exist. Either starvation or apoplectic surfeit would ensue. 



CHAPTER XV. 

Thermal Equilibrium. 

In summing up the general characteristics of mechanical 
energy, in Chapter VI, it was carefully pointed out that all 
energetic conditions of matter varied, not in one direction only, 
from an absolute zero, but in both directions, from a central or 
"mean energetic" condition. It was pointed out in detail how, 
from this central condition, each of the several factors of 
mechanical energy — force, velocity, space and even the mass- 
factor itself — vibrated in either direction indefinitely, yet hmited 
elastically in stable equilibrium. For, as any factor proceeded 
away from the mean condition it engendered forces and phe- 
nomena which tended always to resist its further progress and 
to return it toward centricity. Thus, excessive velocity resulted 
in separation; but separation involved the storage of energy in 
space-form at the expense of velocity-form. Excessive separa- 
tion, on the other hand, annulled the velocity which permitted it 
to exist, and invited the regain of velocity in a return into pro- 
pinquity. 

Thermal Equilibrium. If, now, attention be turned to 
Fig. 12, it will be plain that thermal energy shows every indica- 
tion of following all the characteristics of mechanical energy, in 
its range from the unusually cold and solid conditions of matter, 
as at B, to the unusually hot and fluid conditions, as at H. The 
only difficulty in understanding the fact lies in the necessity of 
comprehending the unusually hot extreme in terms of the 
mechanical phenomena of the earth's surface; which, as has just 
been pointed out, concern themselves with matter in the extreme 
opposite condition, unusually hard and dense. 

In the first place, the range of thermal conditions follows, for 
every substance, some such a curve as BCADEFGH. This curve 
exhibits stability of equilibrium at every point, except where it 
is interrupted by the fields of instability which we call fusion, 
vaporization and chemical dissociation respectively. Across all 
such gaps thermal conditions must jump abruptly. The curve 

182 



THERMAL EQUILIBRIUM 183 

is asymptotic to the axis of absolute zero of temperature XX 
at the left. It can never reach it; and as it approaches it the 
increase in negative entropy, or solidity, becomes increasingly 
great with each step nearer. 

The other limb of the curve, toward H, approaches increas- 
ingly the vertical direction. From knowledge yet available, which 
is comprised in Equation 30, it cannot be said explicitly that this 
limb is asymptotic to a vertical axis. But Equation 30, it must 
be remembered, is based upon too narrow a ground to be extrapo- 
lated into a general principle. It is based, first, upon an assumed 
constancy of the specific heat; yet of specific heats in the higher 
ranges of temperature we possess very meagre knowledge. It 
happens, it is true, that our most recent acquisitions in this 
direction point to an increasing specific heat for gases, as tem- 
peratures rise ; and this would tend to maintain the obliquity of 
the curve at H. But the specific heat of water also rises with 
the temperature; yet this fact is only preliminary to a stage, the 
critical temperature, above which specific heats become very 
much smaller and the isomorphic curve much steeper. 

Finally, reference must be had to the mathematical concept 
of the "perfect" gas, toward whose attributes gases tend as 
they rise in temperature; and this perfect gas, having no vis- 
cosity, could be represented upon Fig. 12 only by a straight 
vertical line. For all of these reasons, taken in connection with 
the fact that every other known energetic function becomes 
asymptotic at its either end, the conclusion cannot be escaped 
that the thermal diagram would also extend its upper end into a 
real, as well as an apparent, asymptosy, if its true form could 
be known ; although it is impossible now to define either its true 
form of function or the lateral distance of its axis from any mean 
thermal condition. 

Of these two axes to which thermal conditions are asymp- 
totic, the first named or horizontal one, hitherto known as that 
of the "absolute zero of temperature," will be referred to here- 
inafter as the axis of absolute solidity of matter, where exist 
no fluidity, no elasticity, no expansivity and no translucence. To 
these characteristics might be added no temperature, infinitely 
negative entropy, no volume, and infinite density. But whereas 
the latter list is meaningless to us, we have an idea that we know 
what the former qualities signify ; for are they not the ordinary 



184 ENERGY 

attributes of solid matter? Yet the method of approach to the 
statement just given was chosen in order to make it clear that 
this axis of absolute solidity defines mathematically a condition 
which can never be reached in nature. Its characteristics are 
those which matter never can exhibit. 

Of the two axes of asymptosy, the other or vertical axis is 
that of the so-called "perfect," or, as it will be called herein- 
after, the absolute, gas. In this condition matter possesses per- 
fect fluidity or no viscosity, perfect elasticity or no disgregation- 
work, and perfect translucence. Incidentally it must possess 
either infinite temperature or infinite volume, or both. Or its 
density must be zero. Viewed mechanically, its orbits must be 
all hyperbolic and none elliptic; its mass must be all satellitic 
and none nuclear. This, too, is a condition which may be 
mathematically defined — and such a mathematically defined limit 
is most useful to the understanding — but it is a condition which, 
however closely approached, may never be reached in nature. 

Now this so-called perfect gas — as if anything which the 
Supreme Intelligence had seen fit to exclude sweepingly from 
the universe could be called "perfect" — is so impossible and un- 
natural a thing that the writer has carefully excluded it from 
all of his teaching. But it has been used so widely, by other 
teachers, as the base and explanation of all thermodynamic 
action, that it must be mentioned here to the extent of putting 
it in its proper place. 

That is to say, there exists on one side of all energetic action 
(at the lower left-hand of Fig. 12) an unattainable limit of 
deficit of internal energy, when all internal motion would be 
purely tangential, or circular, and the pressure zero; which con- 
dition would be attainable, if at all, only when the temperature 
were absolutely zero and the quantity-factor of heat infinitely 
negative; in which condition, if ever attainable, the substance 
would constitute a "perfect" — or, as I prefer to state it, an 
"absolute" — solid. Such a state of affairs would be exhibited 
by the point B, Fig. 12, if pushed far enough to the left (that 
is, to an infinite distance) to bring it into coincidence with the 
axis XX. 

On the other side of all energetic action (but not on the 
opposite side) there exists an unattainable limit of surplus of 
internal energy, when all internal motion would be purely radial, 



THERMAL EQUILIBRIUM 185 

or rectilinear, and none tangential ; and where the viscosity would 
be zero; which condition would be attainable, if at all, only when 
the temperature were infinite and the quantity-factor of heat at 
its maximum, as shown by the axis NM. Such a state of affairs 
would be exhibited by the point H of Fig. 12, if pushed far 
enough up (that is, to an infinite distance) to bring it into coin- 
cidence with the axis NiM to which GH is asymptotic. This 
axis is therefore that of the so-called "perfect" — or, as I prefer 
to call it, the '"absolute" — ^gas. 

Between these two unattainable extremes, or purely mathe- 
matical limits, all natural action, thermodynamic or otherwise, 
occurs — surging back and forth in stable equilibrium about some 
central mean energetic or thermal condition. In so far as it 
gains approach to the vertical axis of zero solidity, or gaseous 
perfection, it gains fluidity, elasticity, expansivity and adaptability 
for work-performance. And in this sense it is true that, to the 
extent to which matter possesses those qualities which, if existing 
alone, would constitute it a '"perfect" gas, it exhibits faculty for 
thermodynamic labority. But equally true is it that in so far as 
it gains approach to the horizontal axis of zero gaseousness, or 
perfection of solidity, it gains faculty for impact, friction and 
gravitational work-performance; in other words — to coin a par- 
allel term — for dynamothermal thermogy. And these two con- 
trasted faculties, it seems to me, are of equal importance and 
deserve equal prominence in thermodynamic discussion. 

Neither limit of thermal attributes can matter ever reach. 
Yet each of these limits has its proper use, as a base of refer- 
ence. Some writers on thermodynamics, if not all, have preferred 
to enter the topic from the limit properly called the "absolute" 
gas as their base of reference. To this plan, so carried out that 
the student sees the true relation between base of reference and 
all natural action, there can be little objection. 

The writer has much preferred, however, to enter the topic 
from the other limit, that of the "absolute" solid, by studying 
first the elements of energetics as they would occur between two 
mass-portions each of which is apparently a solid unit. This is 
natural because, in the first place, the growing boy has dealt 
almost entirely with solids, in work and play, and all his concepts 
of mechanical action are based thereon. In the second place, 
the mechanics of solids is already reduced to an exact mathe- 



186 ENERGY 

matical science, which is supposed to be taught to every student 
of engineering before he meets the problems of thermodynamics. 
So that the writer regards this mode of entrance to thermo- 
dynamics as far preferable to that via the ^'perfect" or "abso- 
lute" gas. 

The one plan which deserves unlimited condemnation, how- 
ever, is to open the study of thermodynamics with the concept 
of the absolute gas, using actual gases as ^'impure" illustrations, 
and to leave the student with the idea that what likeness to the 
absolute gas can be found in nature is alone thermodynamic 
action. All natural action is thermodynamic, to whatever extent 
heat takes part. To leave the student with his concept of natural 
action thus equipped with a roof, in the form of a knowledge of 
the gaseous side of thermodynamics, but with no foundation or 
adequate frame- work in the way of a corresponding knowledge 
of the solid and liquid aspects of the science, is what is being 
done by every teacher who thus misuses the mathematical con- 
cept of the so-called ''perfect" gas.* 

This fundamental fact should be one of the first taught the 
student of natural science. These two axes define the boundaries 

*The following language concerning this situation is taken from 
another article by the writer : "We know of no gas so hot or so rarefied 
and so 'perfect' that it loses all viscosity, or is perfectly elastic and free 
from disgregation-work. We have no reason to think that any such a 
substance could ever exist. It is unobjectionable to refer, upon occasion, 
to the mathematical hypotheses of the 'perfect gas' or the 'perfect solid' 
as aids in argument, as has just been done. But with these references 
should always go the explanation that these two hypotheses constitute 
impossible, supernatural extremes, between which vibrate all known 
natural conditions of matter; and science should have just as little to 
do with the supernatural as possible. 

"But when reliance upon the perfect-gas hypothesis goes so far as 
to make it the center and foundation — the immediate beginning, rather 
than the remote and unattainable horizon — of all thermodynamic study, 
as is now very widely done, the writer feels impelled to rise, in the 
name of nature and common sense, to denounce the practice. His experi- 
ence in teaching thermodynamics has found this method so universally 
confusing to the engineering student, who above all others needs acquaint- 
ance with nature rather than with disembodied and supernatural hy- 
potheses, and so belittling to the dignity of the instructor posing as a 
Guide to the Truth, that he can express his feelings only by a free 
quotation from Mr. Burgess's Turple Cow' — with apologies to the 
Cow, as well as to Mr. Burgess: 

*I never saw a Perfect Gas. 
I never hope to see one. 

But I can tell you, as we pass, 
I'd rather see than be one.' " 



THERMAL EQUILIBRIUM 187 

of the territory within which occurs all natural action. They 
are dead-lines which not only matter, but even thought, may not 
touch without ceasing to be. A great deal of physics may be 
taught, it is true, without mention of either the absolute gas or 
the absolute solid. But if either of them is mentioned at all — 
and their use is common in teaching even elementary physics — 
both should be mentioned together, and both should be displayed 
as supernatural concepts. Here again, neither Siamese twin 
should be presented with the message that the other had not 
yet been separated. It is as absurd to mention to the student 
the absolute zero of temperature without parallel mention of the 
perfect gas, or vice versa, as it is to mention the force of 
gravitation without its inseparable companion, centrifugal force, 
or to describe a chemist's balance as a pan suspended from a 
rod into which substance may be put for weighing, but with no 
mention of the other arm and pan into which the weights are put. 

Indeed, the very description of matter, to the student fit for 
generalities at all, should be made in such a way as to show 
that solidity and expansive fluidity of matter are purely relative 
terms — that all matter is partly solid and partly gaseous, that 
what we call "solid" matter is merely matter more solid than 
gaseous ; that what we call "gas" is merely matter more gaseous 
than solid ; and that nowhere in nature occurs matter which is 
either wholly solid or wholly gaseous. 

Although water alone has been selected for illustration in 
Fig. 12, because of its familiarity in solid, liquid and gaseous 
states, yet these general conclusions as to the characteristics of 
thermal condition and action of mass apply equally to all sorts 
of matter. All that is needed in order to bring the thermal 
characteristics of any substance into the form shown is to alter 
suitably the scales of temperature and entropy. Thus, for the 
hydrogen-oxygen mixture whose curve is shown as hRGH, all 
that is necessary is to expand sufficiently the temperature-scale 
and reduce the entropy-scale, and its field of fusion and vaporiza- 
tion would appear upon the diagram in much the same form as 
that shown for water. Similarly, the curves for any of the 
calcium or silicon compounds, which remain refractory solids at 
all ordinary temperatures, might be brought into similitude to the 
water-steam curve merely by reducing the temperature-scale and 
expanding the entropy-scale. For all known substances pass 



188 ENERGY 

through the solid, Hquid and gaseous states, under suitable con- 
ditions. Fig. 12, representing, as it does, not only these three 
states, but the processes of transition between them, and also 
chemical transformation, may be regarded as displaying the 
thermal action of all known substances — so far as they connect 
with work on the one hand and with chemical energy on the 
other. 

With this in mind, let the curve be traversed from B to H, as 
before, but with thermal conditions treated as micro-mechanical 
ones. 

At B the substance is a "solid." Energetically speaking, a 
solid is a portion of mass all of whose parts act as one, in their 
reaction with external forces. If one portion moves, all the rest 
move simultaneously an equal distance. Between the particles of 
the system there is no motion at all, except tangential motion. 
The tangential motion, however, must be plentiful. 

The necessity for this tangential motion can best be under- 
stood by reference to the story of how St. Patrick cleared Ireland 
of its snakes: Of how he invited all the toads and snakes to a 
banquet, at which no food was provided. Thus the snakes were 
led to devour all the toads, and then to fight and devour each 
other. And the end of all this was that there were left, finally, 
but two snakes, containing all the rest; and then these two each 
got the other's tail in his mouth and swallowed and swallowed, 
until nothing at all was lef t ! 

For the force of gravitation, it is plain, if left to itself, would 
do just this. It would deprive matter of its occupancy of space. 
For we know of no such thing as an absolute or "perfect" 
density. Every dense portion of matter which has yet been 
examined has proven to be merely an association of portions 
still more dense, the further characteristics of which portions 
were as yet obscure to us. Mass and space are apparently inde- 
pendent. The mutual gravitation of mass increases with the 
square of the propinquity, with no known limit. Mass, once 
drawn into propinquity by gravitation, is ever urged on into even 
greater concentration. 

Flere is obviously unstable equilibrium. If gravitation were 
the only cosmic force the universe would quickly become a single 
geometric point, of infinite density. The universe can be imag- 
ined as continuously existent at all only by the presence of 



THERMAL EQUILIBRIUM 189 

universal tangential motion, developing centrifugal force suffi- 
cient to counterbalance gravitation. The more matter concen- 
trates the greater must be these tangential velocities and cen- 
trifugal forces. The greater, too, must be the rigidity of each 
tiny whirling mass-pair. But the student should be cautioned at 
every point against any thought of any absolute or final limit 
to the smallness, density, speed and rigidity of such ''particles" 
of matter. He should be reminded that it is just as foolish and 
needless to speak or think of any ''ultimate" or "indivisible" 
particle of matter as it would be to speak — as the ancients always 
did — of any solid fundament below the universe, upon which it 
rested, or of any rigid limits to space, beyond which the universe 
could not extend and where existed — what? Neither beginnings 
nor ends of anything, whether of time, space, density, solidity, 
elasticity, intensity of energy or aught else, either exist in nature 
or can be comprehended by man. 

The student need not be told this. But his teachers very much 
need to be reminded to refrain from suggesting to him the 
opposite idea. 

Therefore, to return to the cold facts of Fig. 12, however far 
the point of natural condition B may be imagined as departing 
to the left, approaching the condition of absolute solidity more 
and more slowly, it must still be always within a finite distance of 
its mean energetic condition. While all cold, hard solids possess 
very great solidity, rigidity, density and inelasticity, they do not 
altogether lack fluidity, ductility, space, expansivity and elasticity. 
They all occur at very low temperatures, relatively to their liquids 
and gases ; but they always possess some temperature. Heat can 
always be extracted further from them. In short, their condition 
is always transmutable into liquids and gases by processes of 
finite dimensions. We therefore surmise that, while the greater 
portion of the mass of such bodies revolves, in exceedingly dense 
particles, with almost circular motion at exceedingly small radius, 
and with very great rapidity and rigidity, yet there exists not 
only some slight eccentricity to these orbits, but a minor number 
of mass-particles revolve about highly eccentric, or even hyper- 
bolic, orbits. The tiny, rigid, tangential pairs arrange themselves 
into some form of stable equilibrium, called molecular and crys- 
talline. Between and about and out from them shoot the minority 
of satellitic projectiles, exerting some slight expansive vapor- 



190 



ENERGY 



pressure and affording some slight degree of elasticity and some 
slight manifestation of temperature. At human temperatures 
many crystalline "solids" exhibit considerable fluidity, ductility, 
elasticity, etc. ; but when reduced to the temperature of boiling 
hydrogen these same substances become rigid and brittle in the 
extreme. 

It is such "solids" as these which, by their relationship in 
visible distances of separation and visible velocities of motion, 
embody what we call "mechanical" energy. Here on the surface 
of the earth we see these solids related in motions, all of which 
are below the lower critical velocity, and which end promptly 
in collision with the earth and with each other. They are 
therefore inseparably associated with impact and friction ; that is, 
with thermogy. Their energetic history ends always in this. 
Every bit of mechanical energy aroused here on the earth's sur- 
face, whether by our artificial heat-engines, or by the sun-heat 
acting upon wind and water, or by the moon acting upon the 
oceanic tides, ends its existence, in a surprisingly short time, in 
transformation into heat. The region of mechanical energy is 
the region of solidity, opacity, brittleness, inelasticity and density. 




PERFECT SOLIDITY 



clH c 
FIG. 1 2 A. 



OF TEMPERATURE 



THERMAL EQUILIBRIUM. 



191 



It is the region of impact, friction and 
thermogy. There all "work" insists upon 
turning into heat, and heat will not turn 
into work. It is the region displayed at 
the lower left-hand of Fig. 12. It is a 
region which belongs especially to the 
solidified planets of the universe, of 
which our earth is one. It is particu- 
larly the subject of what we call the 
"applied mechanics of engineering" ; for 
while engineering does not deal largely 
with ice, which forms the illustration in 
Fig. 12, yet it does deal, almost exclu- 
sively, with substances, such as iron, 
stone and wood, which are as far below 



DISSOCIATION 








M 



THE REGION OF 

MEAN ENERGETIC 

CONDITIONS 



THE MINIMUM LIMIT OF TEMPERATURE 



FIG. I2B. 



192 ENERGY 

their mean energetic conditions as ice is below boiling water. 

Yet the prime characteristic of this region is, as just stated, 
the thermogic formation of heat. Being the region of deficits 
of heat, entropy and temperature, the prime result of its activities 
is to make good these deficits, in all three lines. Its activities 
being of an extreme energetic nature, they show every disposition 
to get back toward the mean energetic condition. They not only 
tend constantly to produce entropy — which inspection of Fig. 12 
shows that they need even more than temperature — but they 
produce it under such conditions (namely, a deficit of internal 
expansive, and a surplus of external, pressure) that the entropy- 
growth is promptly converted into temperature-rise. Whatever 
causes may be offered in explanation of how these solid bodies 
ever got into this extreme condition portrayed at B, Fig. 12, 
there can be no doubt as to their unwavering tendency to return 
toward centrality, along the path BCADE. 

The impact and friction constantly occurring between such 
solid bodies develops entropy in collision, as has been described, 
by smashing them into finer and finer fragments, until collision 
is no longer possible. Simultaneously, to these fragments are 
imparted sufficient velocities of tangential motion to enable them 
to remain separated fragments, without falling together in obe- 
dience to mutual gravitation. This process we call the creation 
of heat. So long as the substance is still solid, the result of this 
increase in tangential energy is to arouse a resistance from 
without, which compresses the widened circular orbits until 
some portion of the energy is squeezed into a radial, or satellitic, 
form. This increase in radial energy is perceptible from with- 
out, whereas the tangential energy was not, and leads us to 
observe directly that the ''temperature" of the body has risen. 
It is much more indirectly and slowly that we have come to a 
knowledge of the latent, or tangential, increase in entropy which 
preceded this. 

In the case of solids below the fusion-point the internal 
tendency to increase of volume and fluidity, with increase of 
entropy, is resisted by external forces which we are forced to 
call "crystalline," in lieu of fully understanding them. In the 
case of liquids below their boiling-points, however, the internal 
expansive forces are resisted by an external fluid-pressure which 
is familiar to all; and the action of this external pressure, 



THERMAL EQUILIBRIUM 193 

in squeezing the entropy-gains into temperature-gains, is 
exactly that of the piston of the air-compressor or the cush- 
ioning steam-engine, in squeezing volume into pressure and 
temperature. 

When such points as C, or D, or F are reached, however, the 
effects of thermogy, in developing entropy, volume and elasticity, 
is no longer opposed effectively by the external pressure. The 
processes of fusion, vaporization and dissociation, respectively, 
occur in purity, as developments of entropy, latent heat, volume 
and elasticity at constant pressure — as a development of sub- 
division and disgregation of matter, and of radius of tangential 
motion, without any increase in radial kinetic energy. 

It thus becomes plain how matter which possesses a deficit of 
radial energy, of satellitic mass, of volume, expansive pressure 
and elasticity, with a surplus of tangential velocity, density and 
rigidity, begets naturally those things which it lacks and rids 
itself of those things which are in surfeit, as it passes along its 
path of thermal conditions, BCADEFGH. 

It is now to be noted most carefully that the motive power of 
those peculiarities, which started it along this path, dies out as 
it proceeds. That is to say, as it gains radial energy, tem- 
perature and satellitic mass, and more particularly, as it gains 
entropy, volume and elasticity, the effectiveness of impact and 
friction for heat-development die out. The causes which forced 
the substance toward its mean energetic condition lose their force 
as that condition is reached, and counter-tendencies begin to 
prevail. Thermal or other energetic conditions, like pendulum- 
bobs, lose their motive power as they approach their central 
positions. 

This does not occur abruptly or completely. Gradually 
rigidity, density, inelasticity and opacity fade away, but they 
never completely disappear. We know of no substance which 
does not possess these qualities to some slight degree, which 
does not carr)^ on some slight thermogic action. Even the most 
rarefied of gases possess some slight viscosity and absorption. 

With these diminutions in impact and friction must be con- 
sidered also thermal conductivity and the absorption of radia- 
tion. Of these, in detail, we know nothing, except that their 
results are identical with those of impact and friction. Here 
too, in the progress along the thermal path, the likeness again 



194 ENERGY 

appears. Speaking generally, it is the forms of matter which 
are most rigid and dense, and most subject to impact and fric' 
tion, which possess the highest rates of surface-absorption of 
heat and its conductivity throughout the mass. To this rule 
there are many minor exceptions, but they are local and insig- 
nificant. The broad fact is as stated. While the contrast 
between solids and liquids in these matters is not marked, that 
between liquids and gases is so. The gases are most difficult 
to impart heat to. Air is much harder to heat than is saturated 
or slightly superheated steam, and steam is much harder than 
water. As to solids, the difficulty in heating them does not lie 
in any lack of absorptivity, but in bringing ordinary thermal 
media, which are usually liquid or solid themselves, into effective 
contact. Therefore, as matter proceeds along the thermal path 
from the solid extreme toward the gaseous, it loses first its 
liability to impact and friction, and secondly its liability to the 
receipt of entropy by conduction or absorption. 

Moreover, the liability to thermogy does not depend solely 
upon the condition of the body's surface. The rate of heat- 
transfer depends also upon the difference in temperature between 
contributing and receiving body. When the recipient is a very 
cold solid, nearly all other bodies are warmer than it, and very 
much warmer, and contribute heat rapidly. But as the substance 
grows warmer it leaves behind it. one by one, those neighbors 
which were recently able to impart heat to it, and in turn begins 
to impart heat to them. Until finally, when it has become 
unusually hot, as toward the H-extreme of the thermal path, it is 
only the semi-occasional body which is still warmer than it, and 
so able to contribute heat to it. 

At this end of the path, too, rigidity, viscosity, impact and 
friction have almost entirely disappeared. In their place now 
appears an excess of volume, expansive pressure, fluidity, elas- 
ticity and often incandescence. Here work will no longer turn 
into heat, by thermogy, to appreciable degree. Instead, laborify 
has become the prevailing phenomenon. The heat insists upon 
turning into work. And what heat cannot find conditions suit- 
able for its conversion into work, insists upon radiating and 
conducting itself away to colder bodies. 

Viewed constructively, the molecule now appears to consist 
almost entirely of minutely subdivided particles moving in 



THERMAL EQUILIBRIUM 195 

highly eccentric and chiefly hyperboHc orbit. There still exist 
nuclei, but they embody a minor portion of the mass. There 
still exists tangential motion, but it is now an insignificant base 
for the prevailing superpermanency of radial energy. 

It is because of this paucity of tangential motion that it is so 
difficult to impart further heat to the gas, by thermogy ; for 
impact, friction, conduction and the absorption of radiation, all 
must occur through the tangential components of motion, 
although they may find final expression in radial motion. It is 
because of the superabundance of radial energy, on the other 
hand, that it is so easy to raise the temperature of a gaseous 
body by compression; for compression acts directly upon the 
radial component, being transformed into tangential motion only 
secondarily, under the squeezing of the nuclei into greater pro- 
pinquity. Only, it is to be noted, as the temperature of a gas 
increases so also does its pressure, other things being equal ; and 
as the condition of the substance becomes extreme in the H- 
direction it becomes increasingly difficult to find a force capable 
of compressing the gas, although it may become increasingly 
fluid and elastic and capable of compression as it goes. 

Energetic Gravitation. There thus become visible in the 
thermal field two fundamental gravitational tendencies. One of 
these is for all substances, and particularly those to the left of 
the mean energetic condition, to pass horizontally to the right, by 
thermogy. The other is for all substances, but particularly those 
above the mean energetic condition, to pass vertically downward 
in temperature, by labority. Roughly speaking — and perhaps 
accurately too — the tendency to fall in vertical intensity (or, in this 
case, in temperature) is proportional to the intensity itself, or 
the distance from the horizontal axis of absolute zero of tem- 
perature. In saying this we have not in mind the force with 
which it tends to fall, but the chances of that force being able 
to prevail. 

Similarly, the tendency of energy to increase In quantity- 
factor (in this case entropy, although the statement apphes 
broadly to the mass-pairing or quantity-factor of any form of 
energy), by "collision," or resistance encountered In motion, is 
proportional to the distance of the energetic condition from the 
vertical axis of absolute zero of solldltv, or from the condition 
of the perfect gas, at the right. In this we see that we have 



196 ENERGY 

unconsciously formed the habit of speaking of entropy with the 
positive and negative signs reversed from what they naturally 
should be. Just as, in speaking of space-energy, in order to be 
consistent we have been forced to substitute the idea of pro- 
pinquity, or lack of space, for space itself, as a measure of 
intensity, so, in speaking of thermal energy, in order to be 
consistent, we should always refer to entropic changes in terms 
of lack of solidity. Solidity tends to decrease, in the universal 
thermogic aspect of thermal phenomena, just as persistently and 
uniformly as temperature tends to decrease in their work- 
performing or laborious aspects. Had the idea of entropy and 
thermogic tendencies been originally thus handed out to the 
student right end foremost, as a simple universal tendency of 
solidity to disappear in impact and friction, quite similar to the 
tendency of temperature to disappear in work-performance, the 
subject would never have become so enshrouded with mystery 
and awe as it is at present. But now, so strong is habit, it will 
be a long and difficult task before the customary algebraic signs 
of entropy will be reversed. 

These two universal tendencies are the energetic gravita- 
tions. Each prevails as constantly as does the Newtonian cen- 
tripetal gravitation of mass and centrifugal gravitation of motion. 
In mechanical energy these tendencies are, as to its intensity, 
that of either unusual velocity, unusual space or unusual pro- 
pinquity to decrease to a minimum; as to extensity, that of the 
mass-pairing of matter to proceed to a maximum, in its further 
and further subdivision (or, in other words, the tendency of the 
solidity of all matter toward a minimum). In thermal energy 
these tendencies are that of temperature to decrease and that of 
entropy to increase. All thermal phenomena are the result of a 
balance between these two gravitational tendencies ; for the 
gravitations act always in opposition, in a counterbalance between 
each other and with outside conditions. Sometimes one prevails 
over the other, sometimes the other over the one ; sometimes both 
together prevail over outside forces, sometimes both are over- 
come thereby. 

If Fig. 12 be held up by its upper left-hand corner, the 
thermal path BCADEFGH will then appear as the somewhat 
irregular path of a pendulum-bob, swingeing from the point of 
suspension. Just as the bob of a real pendulum is impelled 



THERMAL EQUILIBRIUM 197 

toward its central position from either side, whichever it may 
chance to reach, so is the thermal condition of matter impelled 
always toward its mean energetic condition, from either thermal 
extreme which it may chance to attain, by these two vast gravita- 
tional tendencies. Across this great arc of physical condition 
all thermal phenomena are constantly swinging back and forth, 
as in a gigantic pendulum. For the ranges of the arc are indeed 
gigantic. From the coldest density of solid matter lost to view 
in interstellar space, on the one hand, to the highest temperatures 
of the incandescent suns and the extreme tenuity of the luminous 
nebulae and comets' tails, on the other, thermal happenings are 
constantly swinging. And no heat-action on the surface of the 
earth can be understood without reference to both of these 
extremes. 

What combinations of conditions in nature's vast laboratory 
may have led to the existence of suns and nebulae on the one 
hand, and of remote dark stars and cold meteorites on the 
other, is a great and open question. Its answer has nothing 
to do with the present argument. The fact remains unques- 
tionably fact that, these extremes of heat-condition once in 
existence, their tendencies must be as described. 

Yet the conclusions therefrom must not be too hasty. Carnot 
and others before him noticed the downward tendency of tem- 
perature. Clausius noted the outward tendency of entropy — or 
the downward tendency of solidity. It was Zeuner who deduced 
therefrom the quite unwarranted conclusion that the entropy 
of the world was not only tending, but was actually moving, 
toward a maximum. It was Lord Kelvin who put these ideas 
together into the doctrine of the steady "degradation" and loss 
of availability of energy. 

These mistaken conclusions depend upon seeing these two 
tendencies as working always and everywhere together to a single 
end, viz : to the reduction of all energy to heat, and all heat to a 
maximum of entropy and a zero of temperature-difference. 
The facts are just the opposite of these. The two tendencies 
are always opposed. Entropy cannot be increased unduly except 
by processes which simultaneously increase the intensity of tem- 
perature and its availability for labority. Therefore entropy 
itself constitutes a factor in availability. For thermal radiation 
and work-performance it becomes available when extreme in 



198 ENERGY 

the positive direction. For thermogy and mechanical energy it 
becomes available when in an extreme condition which we, mis- 
takenly, call ''negative" (as at B, Fig. 12). For matter in this 
extreme condition embodies as great an intensity, or availability, 
of mechanical energy as that in the extreme condition at H 
embodies intensity and availability of thermal energy. Indeed, 
the B conditions are attained by matter only when it becomes 
so remotely isolated from its fellows, in the heavens, that any 
occurrence of thermogic collision at all must develop so much 
heat as to transfer instantaneously all the matter involved into 
an equally extreme condition of temperature, as at H. 

The right or wrong of these ultimate conclusions of the 
physicist is of little importance to the engineer; but a correct 
understanding of the familiar thermal phenomena upon which 
they are based is so. It is therefore worth while to state how 
the doctrine of degradation — as commonly taught, if not as Lord 
Kelvin stated it — is inconsistent with the natural facts. 

This doctrine, stated briefly, says that the hot half of the 
universe, consisting chiefly of the suns, is steadily becoming 
colder, by radiating its heat to the colder half, which latter is 
steadily becoming warmer by absorbing it. Therefore all tem- 
perature-differences, and with them all availability of heat for 
work, must ultimately die out. The fallacy here is twofold, and 
both its aspects are obvious to any technical graduate. 

In the first place, which is the cold half of the universe? 
What is the average temperature of the material world? Cer- 
tainly far above any temperatures familiar upon the earth's 
surface. The mean temperature of the earth itself, considering 
its interior, must be above 2,000° F. Yet it is a cold planet. 
The planet Jupiter, exceeding all the other planets together in 
mass, possesses a low red heat even at its surface. The sun, of 
enormous mass, several hundred times that of all the planets 
together, possesses a mean temperature certainly upwards of 
10,000° F. Unquestionably the mean temperature of all the 
solid matter of the universe — using the term solid here to signify 
all mass-portions which can be distinguished as separate units, 
like the stars, etc. — is a white heat. It is only the minute frag- 
ments, scattered occasionally throughout space, remote from the 
incandescent centers of congregated mass, or suns, which are ever 
"solid" in the sense which we use in engineering. 



THERMAL EQUILIBRIUM 199 

Professor Poynting has written beautifully concerning the 
temperature-equilibrium of matter, showing how the external 
temperature of any celestial body is the result of a balance 
between absorption and radiation. He shows that it is only as 
we inspect the smaller and smaller bodies, more and more remote 
from the incandescent centers, that the lower temperatures are 
found. The temperature of soHd bodies is determined almost 
solely by radiation — quite as it is with the articles in a room 
containing a fire. Those nearest the fire are the warmest, and 
those most remote are the coolest. As heat radiates from the 
incandescent suns of the heavens it penetrates regions which 
become colder and colder as it goes. Sooner or later it must 
all be intercepted and absorbed. But there is no lower hmit of 
temperature beyond which radiation will not extend, provided 
it chances to escape absorption long enough. It is only the mean 
distance of separation between solid bodies in interstellar space 
which determines the mean lower limit of temperature to which 
radiation falls, before it experiences arrest and absorption. 

Temperature, then, is not a function of time, except in a 
minor fashion, but primarily of separation. It is only as mass 
is separated into remote fragments that it can become cold, hard 
and solid. Time, it is true, introduces a lag into the adjustment, 
so that departing bodies are always hotter than the temperature 
of equilibrium, while approaching ones are colder; but this is 
merely incidental. 

The earth, then, belongs decidedly to the cold half of the 
universe. If so, we should have observed here, according to the 
degradation-theory, a steady accumulation, rather than degrada- 
tion, of temperature. Yet it is the obvious opposite of this which 
form.s the main support for the degradation-hypothesis ! 

In the second place, the doctrine of degradation imphes the 
existence, somewhere in the universe, of an immense mass of 
matter which has somehow become cold enough, and is large 
enough, to be capable of absorbing all the radiation from the 
millions of suns already known to human astronomy — not to 
mention the fact that we are discovering more suns every day, 
almost as fast as we can count and catalog them. The amount 
of matter requisite to do this must be enormous — far greater 
than the mass of all the suns together — and it must be extremely 
cold. For there is a rigid lower limit to temperature-range, the 



200 ENERGY 

absolute zero, so that matter may be warmed up by only a 
thousand degrees or so before it loses its solidity ; but above 
there is an unlimited range of temperature through which radiat- 
ing matter may cool itself down. Any engineer who has handled 
steam-condensers will appreciate the task of finding a proper 
cooling-medium for absorbing all the sun-heat of the heavens, 
without danger of overheating the medium. 

There is ample interstellar space to contain all this matter — 
though there might be an awkward question why it did not 
interfere more obviously with light-transmission through space. 
But the unanswerable question is : How did this matter ever get 
so cold? It could not have done it by expansive labority ; for 
that is a process confined to the gaseous substances, and could 
never reduce a substance to a solid, nor even make any approach 
thereto. Labority tends to result in a low-temperature rarefied 
gas, and might account for the formation of cold attenuated 
nebulae, but never of solid planets. 

As for cooling by radiation, or thermogy, it is inconceivable 
that any body could ever so cool itself in that way as to turn 
around again and, in the same locality, proceed to absorb heat; 
and that at enormous rates, in enormous quantities. It is quite 
imaginable that a body should so cool by radiation that its rate 
of radiation should become almost zero. The cold bodies of 
remote interstellar space are in this condition. But it is incon- 
ceivable that its rate should ever squarely reach zero, and still 
less that it might ever develop a deficit of temperature, against 
its own radiative tendencies. 

The fact is that radiation is everywhere cooling matter. 
Thermal energy never tends in any other direction than down- 
ward in temperature. It is the very universality of radiation 
and temperature-drop in heat which denies the doctrine of 
degradation. 

Yet, in spite of this universal tendency of heat down- 
temperature and the resultant inconceivably vast flood of radia- 
tion pouring forth from every sun into the furthermost crevices 
of space, the mean temperature of the universe remains constant. 
Temperature is being recovered as fast as it is lost. Thermally, 
it never returns ; but mechanically it does. As heat, it flows 
only downward in temperature. But it finds chance to flow 
down-temperature only as it flows outwardly into space; and 



THERMAL EQUILIBRIUM 201 

in remote space it can find embodiment only in exceedingly 
remote, rigid and inelastic solids. But in such solids is also 
embodied the quintessence of mechanical energy. The intensity 
and availability which has been lost to thermal forms has been 
regained in mechanical form. 

Ultimately these remote, cold, hard solids must end their 
existence in collision, r^ulting in gasification, expansion and 
incandescence. The swing of the pendulum will have been 
reversed. The intensity and availability of mechanical energy 
which they embodied will have been lost, and in its place will 
reappear availability for radiation and elastic work-performance. 

Thus there exist always and everywhere two great thermal 
tendencies, which are balanced against each other in cosmic 
equilibrium : 

First: All heat tends always and everywhere to fall down- 
temperature, either by radiation or by work-performance. 
Nowhere in the universe has ever been observed any cessation 
or reversal of this tendency. Usually, the vast flood of radiant 
energy pouring outward into space from all the countless 
millions of suns finds its chance for direct radiation into remote 
space. This is true of the bulk of all sun-radiation ; and in 
this form traveling outwardly at the inconceivable rate of 
i86,ocx) miles per second, it may exist for eons. Light is as 
permanent a form of energy as mechanical energy or heat. 
The universe maintains its stock of it as permanently as it does 
one of the space or motion-energy of celestial bodies.* 



*0f the radiance emitted by our own sun, for instance, about one hun- 
dred millionth is arrested and degraded before the confines of its own 
system are reached and passed. Beyond those confines, apparently, is 
nothing to arrest it until the next solar system is reached. When that 
occurs, assuming the similarity of all solar systems to our own, wherein 
about one-fiftieth of all the surface presented is dark, about one-fiftieth 
of the radiation arrested would be degraded, by collosion with opaque 
bodies. The other forty-nine fiftieths would be arrested by the central 
sun and reflected without degradation. 

But even these proportions are divisions of what is itself but a minute 
fraction of the whole. Assuming that the mean inter-stellar distance is 
but eleven light-years, the proportion of the entire radiation arrested by 
each solar system would be only a decimal fraction of the whole consist- 
ing of unity at the seventeenth decimal place. In other words, it would 
not be until 100,000,000,000,000,000 solar systems had been met, after 
II X 10" years had elapsed, that all of the original radiation from our 
.sun would have ceased its continuous existence, and been either reflected 



202 ENERGY 

In small part, however, this great current of radiation finds 
itself intercepted by a planet equipped with air and water. It 
becomes entangled in the tasks of raising ocean-water into 
clouds, or rearing lofty forest-trees for making coal-beds, or 
driving reluctant engine-pistons before it as it goes. Imprisoned 
in the latent form of lifted weights or stored hydraulic reser- 
voirs, or of chemical energy of wood, coal, oil or gas, it may 
tarry shortly here on earth before it proceeds. But only briefly. 
Sooner or later it has regained its thermal form and liberty 
and is off again, outward into space and downward in tem- 
perature, never to return voluntarily an inch or a degree, as 
heat. Beaten back up-temperature temporarily it may be, by 
superior force, as in our air-compressors and refrige rating- 
machines ; but it always resists stubbornly and may be confined 
only temporarily. Soon it eludes us again, and is off for the 
frightful abysses of interstellar cold and darkness — to relieve 
them as it may. 

Secondly: All lack of heat (or solidity) tends always and 
everywhere to increase in entropy, by impact or friction, or to 
decrease its embodiment in solidity. When the wandering radia- 
tion finally reaches its destination and is absorbed by some 
remote solid, the lower the temperature of its new home the 
greater must be the latter's separation from the great mass- 
center which did the radiating, and the greater must be its 
rigidity and inelasticity. Always and everywhere such remote 
hard solids tend to fall toward the greater aggregations of mass. 
This tendency is just as incessant as is that of heat downward 
in temperature and outward in space. So remote and hard are 
most of them, and so intense is their kinetic energy when they 
do fall, that they are fit only for the manufacture of incandescent 
suns. But some are quite near the surface of the earth, stored 
in the hills, and upon their way toward greater propinquity they 
too may stop a moment to perform work for us — driving our 
water-wheels, accelerating our railway-trains on down-grade, or, 
as in the case of the moon, cleansing our coasts with tidal flow. 

But, like the heat, we can arrest these only temporarily. The 

or degraded ; and since, at each star, forty-nine fiftieths of this would have 
been reflected without degradation, it is not until fifty times the above 
number of solar systems should have been met, after 55 X 10'^^ years, that 
all of the original radiation would have suffered true degradation. 



THERMAL EQUILIBRIUlVI 203 

tendency is all in one direction, and is irresistible. The sand 
washed down from the hills never returns. Each year the earth 
is a smaller sphere than before, and a colder, harder one. 
Nothing will ever recover this ground until the earth's final 
collision with some dark star ends its history as an earth, and 
begins the story of a new sun and solar system of habitable 
planets. 

Equilibrium between Interchangeable Forms of Energy. 

Thus, just as it has been found that within mechanical energy, 
and so also within heat, if heat be a ''mode of motion," there 
exists an eternal equilibrium between spacial and kinetic energies, 
so exists also between mechanical and thermal energies an eternal 
equilibrium of two great opposed tendencies, or gravitations. 
Heat, as heat, tends always only in one direction : downward in 
temperature, turning into work on the way, if it must. Work, as 
work, tends always only in one direction : downward in velocity, 
propinquity and solidity, turning into heat as it goes. Tempo- 
rarily and locally either tendency may overcome the other; but 
universally they are perfectly balanced. The mean availability 
of each form of energy remains eternally constant. 

The Rejuvenation of Intensities of Energy. Thus, while 
it is an invariable law that the tendency of either heat or mechan- 
ical energy must be downward in intensity, so long as it retains 
its original form, yet it is an equally invariable law that, sooner 
or later, this downward tendency must result in transformation 
of the energy. When the energy takes its new form the intensity 
also takes a new form. Herein appears a most important fact, 
viz : the degree of intensity of the nezv form of energy is inde- 
pendent of that of the old. 

Just how far to urge the accuracy of this statement the 
writer is uncertain. It might almost be said that, as far as the 
old intensity was below the mean energetic condition, the new 
intensity must be equally above mean intensity. Apparently this 
is the only true and consistent statement; but, because we lack a 
perfect means of comparing intensities of different forms, the 
writer prefers to make the statement tentatively, until he can 
pursue the question more at length. 

This much is obvious, however, that when the energy has 
undergone a second transformation, back into its original form. 



204 ENERGY 

the intensity then takes a form which can be compared accu- 
rately with its degree before the first transformation. It is now 
obvious that the new intensity of the original form, rejuvenated 
by having undergone a double transformation, is quite inde- 
pendent of the original intensity. The final intensity is deter- 
mined only by external conditions which, for the present, may 
be relegated to the convenient term chance. 

This fact forms the basis for the statement of a general law 
of Gravitation of Intensities of Energy, as follows : 

Energy tends ever, so long as it undergoes no transformation, 
to gravitate to a lower degree of intensity. This tendency ceases 
only with transformation. The energy of a mass-system can 
never regain intensity except by two methods: (i) by a con- 
tribution of energy from some external mass-system, or (2) by 
undergoing a double energy-transformation, into some other 
energy-form and back again."^ 

In order to determine the extent to which this law applies to 
extensities of energy, as well as to intensities, we should have 
to go further into the relationship between different energy-forms 
than seems profitable here. To show where this question leads, 
it may be pointed out that the downward gravitation of mechan- 
ical mtensities, tending always toward collision, impact, friction 
and thermogy, is equivalent to a similar gravitation of thermal 
(?A'tensity, toward an increase in entropy, or a decrease in 
solidity. This fact has already been recognized in the statement 
that heat possessed two gravitational tendencies, one downward 
in temperature and the other outward in entropy, or downward 
in solidity. It now becomes clear that, of the two energetic 
tendencies possessed by each form of energy, one is identical 
with one of the two belonging to a strange form of energy on 
the one hand ; the other is identical with one of the two tendencies 
possessed by a strange form on the other hand. To trace this 
interweaving of energy-forms in exact language is beyond possi- 
bility at present; but this correlation of intensities of different 
and contrasted energy-forms must be mentioned, as essential to 
the principle of universal energetic equilibrium. 

*This law the writer has been teaching since 1899, or possibly 1898. 
In TQOO it was written into the MS. of his "Thermodynamics of Heat- 
engines," appearing in the first edition of that book. These dates are from 
memory only. 



THERMAL EQUILIBRIUM 205 

The Second Law of Thermodjmamics. This general law 
of energetic gravitation just stated has been represented,, in the 
published literature of thermodynamics, by the so-called "Second 
Law." This latter confines itself solely to the downward tenden- 
cies of temperature alone. The present aim is to say merely 
enough to show that the law regarding temperature-fall is but a 
narrow and special expression of a fundamental energetic prin- 
ciple, which runs through all known forms of energy. 

It should also be added that the absurdity of calling this 
merely thermal, special and partial aspect of a great principle the 
"Second" fundamental law has now become apparent. The 
engineering world is now far too frequently occupied with 
energy-transformations to countenance longer the use of a special 
nomenclature for each different form of energy, independent of 
and inconsistent with all the others. We cannot tolerate one set 
of laws for mechanics, another for heat, a third for chemical 
action and a fourth for electricity, when all four of these sorts 
of the same energy are at work in nearly every engine-room in 
the country. The laws of energetics must be codified as such, 
covering all special forms of energy. Should convenience then 
dictate the use of a specialized version of one or more of these 
laws, in any particular field of engineering, it would be well 
enough. But the foundation should be broad and secure. 

Fortunately for this, the order of importance of the basic laws 
of energetics appears to coincide with the order of their chrono- 
logical appearance. The Conservation of Mass was discovered 
first, the Conservation of Energy second, and the Conservation 
of the other Factor of energy than mass — whether it be called 
Motion, Space, Propinquity, Temperature, Intensity or what you 
please — came last. This apparently reserves the place of Fourth, 
rather than "Second," for the Law of Gravitation of Intensities, 
which is distinctly a secondary, rather than a primary, principle. 

The Summation of Energetic Intensities. The true ener- 
getic intensity of any mass-system is in reality the sum of the 
intensities of its several forms of energy. Its thermal intensity, 
for instance, may become very low by reason of its mechanical 
intensity becoming very high, and vice versa. But, according to 
the law of the conservation of energy, the sum must tend to 
remain constant. Where the chances of environment throw the 
bulk of this total intensity into any one form, there may appear 



206 ENERGY 

to be a creation or a loss of intensity; but it is only apparent, 
not real. 

Every mass large enough to constitute a radiant center is 
constantly attracting to itself quantities of matter so cold and 
solid that their impact contributes enormous funds of heat. 
Even the earth is said to collect some twenty million meteorites 
daily. In these collisions occurs an example of a basic ener- 
getic phenomenon, the summation of intensities. The sun, for 
instance, may be taken as the hottest known body. Yet however 
hot it may be, it can never be hot enough to prevent a body 
which falls into it from increasing its energetic intensity. It 
may be so hot, it is true, and therefore so gaseous and elastic, 
that the falling body cannot make it perceptibly hotter. But it 
can and must increase its gaseous volume; and the intensity of 
spacial energy thus stored will abide potentially, without loss 
with time, to make good the temperature-drop which would 
otherwise occur as radiation proceeds. Incandescent radiation, 
like every other energetic process, cannot assume a too great 
intensity without producing conditions which limit its further 
increase. 

It now becomes plain, too, why our heat-engines and other 
machines always have so poor an efficiency. It is because we 
are confined, on the surface of the earth, to a locality peculiar 
in being below the mean temperature of the universe. If we 
could only build our machines of nebular gases, instead of from 
wood and steel, and could jacket them with incandescence, we 
should soon cease our complaints of poor mechanical or thermo- 
dynamic efficiency. Indeed, if we were such salamanders as to 
be able to live where such procedure were natural, we should 
no longer prize temperature and motion as we do now. Instead 
of struggling ever to secure small supplies of super-temperature, 
to warm our bones and run our engines, and then struggling 
further to convert a fraction of this into much prized motion — 
only to have both heat and motion leak away promptly into 
dissipation — we should then seek everywhere for that rarest of 
all things : a chill and a bit of solid fixity. Everywhere would 
be heat. Everywhere would be motion. Flames, whirlwinds 
and hurricanes of gigantic dimensions would overwhelm us at 
every hand. Only rarely, and as a great prize, might we find a 
morsel of peace and quiet coolness, as a firm foundation for our 



THERMAL EQUILIBRIUM 207 

salamandric purposes. But in another instant that too would 
be melted, vaporized and swept away from our grasp into dis- 
sipation, by the universal surplus of heat and motion. We 
should then appreciate as facts what now our wits ought to 
teach us, viz: that cold is just as much a promoter of cycles, and 
is just as valuable to nature, as is heat; that solids and motion 
are inimical phenomena ; and that happiness does not consist in 
always having our own way. 

These facts we appreciate already, in a vague, empirical way, 
if we do not teach them. A machine built entirely of solids we 
know to be inefficient; so we insert fluid lubricating-oil between 
the impinging solids. Or we put our solids afloat, for efficient 
motion, and use an ocean of sea-water as a lubricant. Best of 
all, we already appreciate the efficiency of utilizing the gaseous 
atmosphere as a lubricant, between our rapidly flying air-ships 
and the earth. 



CHAPTER XVI. 

Transformations and Conservations. 

When attention is turned to other forms of energy than work 
and heat, while questions as to their exact structure become 
more and more obscure and uncertain, yet their mutual identity 
with heat and work, in all of their fundamental characteristics, 
becomes more clear. While in the case of work it proved to be 
possible to analyse all features with exactness (except that no 
expression for the fund of tangential energy could be found), 
when heat was reached it became necessary to deal merely with 
general attributes, averaging for numbers of component mass- 
portions too great for individual treatment, and reserving elas- 
ticity of definition to cover conditions impossible of definition. 

As the discussion proceeds from work and heat to the other 
forms of energy, the haziness of ideas as to exact structure — at 
least, when treated by the writer — must extend rapidly. It is 
surprising, indeed, that even an apparent identity may be dis- 
cerned. Yet careful examination reveals considerable ground 
common to all the energy-forms. 

First, all the known forms of energy are mutually trans- 
formable. Some part of any fund of any form of energy is 
always capable, under favorable conditions, of transformation 
into any other form; and in every case the Conservation of 
Energy holds true. Thus, if to the list of energy-forms already 
discussed there be added electricity, radiant energy (light), and 
biological (animal or vegetable) energy, instances of their mutual 
transformation familiar to the student can be found between 
each two, with one or two possible exceptions. This leaves out, 
among the familiar energy- forms, only sound ; and the amount 
of energy involved in most audible phenomena is too small for 
perception after transformation into the other forms listed. 

208 



TRANSFORMATIONS AND CONSERVATIONS 209 

These instances of mutual transformation might be Hsted as 
follows : 

Thermal to Mechanical : Expansion under heat; the steam- 
engine. 
Thermal to Chemical : Dissociation ; the lime-kiln. 

Thermal to Electrical : The electropile. 

Thermal to Radiant : Incandescence ; all flames and lamps. . 

Thermal to Biological : The direct effect of sun-heat upon veg- 
etable and animal life. 

Mechanical to Thermal : Impact and friction ; compression. 

Mechanical to Chemical : Detonation. 

Mechanical to Electrical : The dynamo or glass-plate machine. 

Mechanical to Radiant : (None known.) 

Mechanical to Biological: (None known — unless the stimulative 

effect of the slipper or the shingle be 
admitted to scientific dignity.) 

Chemical to Mechanical : The chemical fire-engine. 

Chemical to Thermal : Combustion. 

Chemical to Electrical : Primary and secondary batteries, dis- 
charging. 

Chemical to Radiant : Phosphorescence. 

Chemical to Biological : The consumption of animal tissue. - 

Electrical to Mechanical : The electric motor. 

Electrical to Thermal : Electrical resistance. 

Electrical to Chemical : Secondary batteries undergoing charge. 

Electrical to Radiant : Crookes tubes. 

Electrical to Biological : Galvanism in medicine. 

Biological to Mechanical : Animal activity. 

Biological to Thermal : Animal heat. 

Biological to Chemical : The accumulation of fat and tissue. 

Biological to Radiant : Animal phosphorescence ; the glow- 
worm. 

Biological to Electrical : Animal electricity ; the electric eel. 

This array of energy-transformations is most Impressive and 
significant. While there are two combinations in the above list 
for which no instance is known to the writer, and while sound 
and magnetism would have to be added to the list to make it 
complete for the inanimate energies of the earth's surface, with 
celestial and sociological energies on beyond in either direction, 



210 ENERGY 

yet virtually it may be said that there is known to man an 
instance of mutual transformability, in either direction, between 
every two known forms of energy. Some of these phenomena 
are rare and obscure in occurrence, while some are of every-day 
famiHarity. All in all, they cover an exceedingly diverse field 
of intricate natural action. 

It is next to be noted that the identity of heat as a mode of 
mechanical motion originated from and rested upon, until 
recently, nothing more than this mutual transformability. When 
the mechanical equivalent of heat was once determined exactly, 
the identity of their natures was considered proven. Yet it is 
merely because the transformations between heat and work are 
so much more familiar than those between the other forms — or 
perhaps because they are the most familiar of those capable of 
exact definition, which electrical, radiant and biological energy 
are not — that their identity was foreshadowed so long ago and 
is now adopted with so little question. It was more than two 
centuries ago that Robert Boyle intimated the identity between 
heat and work. It is more than one century since Count 
Rumford proved the fact qualitatively, and almost quantitatively, 
upon a large scale. It is almost half a century since Joule proved 
it quantitatively and exactly. And yet, in this entire history of 
the progress of the idea of the identity between heat with work, 
not one bit of evidence has been adduced which is more con- 
clusive than the universal mutual transformability of the two. 

But a belief in mutual identity thus supported must extend 
as far as does mutual transformability, viz : to any and all com- 
binations of energ}^-forms. Such evidence has to-day accumu- 
lated, for chemical and electrical energies, far beyond that exist- 
ent for work and heat in either Boyle's or Rumford's time, if 
not also in Joule's. Their equivalence has been brought into the 
circle with the accuracy which was attained for heat only in 
Joule's hands. As to radiant, biological, sociological and celestial 
energies, no units for their quantitative measurement exist, and 
therefore coefficients of equivalence are of course impossible ; 
but the instances of their mutual transformability, apparently 
with quantitative conservation, are multiplying daily in familiar 
experience. 

It is also to be noted that even those coefficients of equiva- 
lence which have already been determined rest upon transforma- 



TRANSFORMATIONS AND CONSERVATIONS 211 

tions in only one direction. Rumford's and Joule's work was all 
done in the production of heat from work. No fixed equivalence 
in the production of work from heat has ever been sought or 
found. The applicability of Joule's equivalent to work-perform- 
ance by heat is pure assumption, checked only by indirect evi- 
dence as to the correctness of Carnot's law. The same is true 
of thermo-electric equivalence. We know that a definite amount 
of electricity will produce a certain amount of heat, but not that 
an equivalent amount of heat will produce its- proper quota of 
electricity. 

The question as to the identity of all these various forms of 
energy, as all being modes of motion and forms of separation 
between mass-particles, and as all being amenable to the laws of 
energetics, lies therefore in this state of settlement: That the 
evidence in favor of the identity of work, heat and chemical 
energy is so overwhelmingly great that no one to-day dares to 
deny it, or to suggest an adequate substitute hypothesis ; yet that 
there exists, in the face of an only slightly less abundance of the 
same form of evidence, an astounding reluctance to admit the 
same identity between other forms. 

Of these, electricity occupies an intermediate position. Static 
electricity has been positively identified as a matter of mass ; but 
concerning kinetic electricity we are still forced to rely upon 
inference. Light and magnetism are still open to the most varied 
hypotheses. 

The Universal Interchangeability of Energetic Form. 
Yet mass is merely the only accurate measure for quantity of 
matter. The next best definition of mass is as that which exerts 
force or stress, or exhibits strain under stress. Without the 
concept of mass the words "strain" or "stress" or "wave" or 
"pulse" lose all significance. It therefore seems to the writer 
that those who insist upon the doctrine that the ether, for 
instance, is not massive are just as reckless with language as 
were the earlier writers upon phlogiston or the degradation of 
energy. What a "strain or pulse in the ether" may be, if it does 
not signify the dislocation or acceleration of mass, the writer 
cannot imagine. The writers who use words to describe etheric 
action which imply naught but a reference to familiar mechanical 
(that is, massive) phenomena, and who yet simultaneously dis- 
claim any such parallelism, cannot appear otherwise than as 



212 ENERGY 

tangling themselves in words — albeit as artistically as a Laocoon. 

Even the late Lord Kelvin, in his 1907 paper before the 
British Association, assumes that the ether has no mass. Yet 
he speaks of a "pulse" or ''disturbance" of the ether, and then 
assumes without argument that this disturbance involves energy. 
But if the ether is massless its disturbance would not necessarily, 
nor even probably, involve energy. The words ''disturbance" or 
"wave" carry an energetic significance only when the disturbance 
is that of mass. For the only known thing which absorbs energy 
in its disturbance is mass. The ether can form no exception to 
this statement, for the ether is unknown. 

Again, he says : "There is no difficulty in this conception of 
an utterly homogeneous elastic solid" (the ether). There is no 
difficulty in the concept of the ether as a solid; but there is as a 
homogeneous elastic solid. The only energetic systems man has 
been able to dissect, with mathematical accuracy, are the celestial 
systems ; and in these the only instance of either elasticity or 
energy occurs as a function of heterogeneity — of a dissociation 
and interaction at a distance between two or more bodies which 
may be quite dissimilar, and each of which may be totally 
inelastic, but both of which are massive and are in motion. This 
elastic action at a distance is quite compatible with the "solidity" 
of the pair, by any criterion for solidity except homogeneity. 
The "homogeneous elastic solid" is much like Voltaire's famous 
"Holy Roman Empire," which turned out to be, upon inspection, 
neither holy, nor Roman, nor an empire. That is to say, we 
can have no concept of a truly homogeneous substance as pos- 
sessing any qualities whatever; but the one especially complete 
lack-of-quality which a homogeneous body must have is absolute 
inelasticity and passivity. 

The same looseness of thought and diction prevails in the 
frequent reference to an "indivisible unit" of matter. For a 
thousand centuries man has existed in ignorance of any portion 
of matter smaller than the tangible or visible. During the last 
century of this thousand he has confidently regarded the chemical 
atom as the ultimate indivisible unit — although not a bit of the 
evidence upholding the atomic theory indicated that the atom, 
however small a division of matter, was the ultimate subdivision. 
Nevertheless, during the last thousand days or so he has readily 
accepted the idea of the electron, a thousand times smaller than 



TRANSFORMATIONS AND CONSERVATIONS 213 

the atom. Yet he now stands, apparently, just as firmly as ever 
for the idea that the electron is now really the ultimate unit of 
matter — although he already knows of three different sorts of 
electron, implying an internal configuration, and consequently a 
relation of parts. 

The sensible man of the educational profession, and the 
sensible undergraduates as well, resent these childish incon- 
sistencies. They know well that another thousand days may see 
the proof that the electron is itself divisible into very many still 
smaller mass-units, as yet undiscerned. They can see, both 
rationally and instinctively, that, whatever may be the last dis- 
covered smallest portion of mass, its energetic activity must 
imply at least some degree of elasticity, and therefore of hetero- 
geneity of structure, held apart kinetically. They know that the 
one thing unknown to nature or the laboratory is homogeneity; 
that everything proves, upon dissection, to consist of a mere 
heterogeneous relationship, between parts which individually 
possess none of the features evinced by the relationship. The 
fact that many things, perhaps familiar things, yet remain undis- 
sected is no evidence whatever in negation of this idea. 

For this great question of the identity between the several 
forms of energy no conclusive evidence can be expected. There 
can be brought to bear upon it, however, a second line of indirect 
evidence. This rests upon the identity of all these diverse 
energy-forms in their constitution and their characteristics of 
action. 

The Universal Dualism of Dimension in Energetics. 
Early in the study of mechanical energy it was pointed out 
that this best known of all energy-forms is composed of two 
factors, intensity and extensity. These two factors combine, 
not as a sum, but as a product. Neither factor may vary toward 
zero except as the other factor varies toward infinity. There- 
fore, in spite of the unceasing gravitation of each factor toward 
its own zero, neither may travel in that direction with other than 
negative acceleration and increasing resistance, because it is 
thereby forcing the other away from its zero with positive 
acceleration. 

These same characteristics apply to all the more obscure 
forms of energy, with fair completeness. Each form of energy 



214 



ENERGY 



possesses a dual nature, consisting of two dimensions or factors, 
its own particular forms of intensity and extensity respectively. 
The gravitational tendency of each of these factors may be 
observed. 

This duality of nature and identity between the several 
factors may be indicated by the following table :* 



Form of 


Factor of 


Intensity : 


Factor of Extensity : 


Energy : 


Name 


Unit 


Name 


Unit 


Mechanical: 
Potential: 
Approx. : 
Exact: 


Distance 
Propinquity 
1 


Feet 


Force 
Mass-squared 

Mass 
(Mass)2 

Charge 
Current 

Mass 

Entropy 
Entropy 


Pounds 
c (Lbs.-^G^2 




Feet-per-sec. 
(Feet-per-sec.)2 

Lbs.-f-G 

Volt 
Volt 


Kinetic: 
Approx. : 

Exact: 

Electrical: 
Potential: 
Kinetic: 

Chemical: 
Potential: 


^ distance ^ 

Velocity 
(Velocity)" 

Total mass 

Potential 
Potential 


Lbs.-^G 
(Lbs.-7-G)2 

Coulomb 
Ampere 

Molecular wt. 


Thermal: 
Kinetic • 


Temperature 
Disgregation 


Degree (abs.) 


B. t. u. 


Potential: 


Abs. Temp. 
it 



Although this table is not complete, the correspondence 
between the several forms of energy, in the possession of two 
factors or dimensions of energy, one of which is probably a 
function of motion, distance or force and the other of mass and 
its subdivision, is obvious. In mechanical energy the items are 
complete; except that we lack a unit of measurement, or even 
the familiar concept, of propinquity as the intensity-factor of 
potential energy; and we lack an exact expression for the tan- 
gential or latent fund of mechanical energy. 



*The writer wishes to record here the statement, although It is impossi- 
ble to support it here, that what study he has made of social and eco- 
nomic energies reveals the same duality of form running through them 
all. Some of the material for this appears in his "Cost of Competition." 
Much yet awaits development. Of the duality of factors, however, there 
can no longer be any doubt. 



TRANSFORMATIONS AND CONSERVATIONS 215 

In electrical energy all the items are complete; but only in 
the case of static electricity has it been settled that the extensity- 
factor is identical with mass. The intensity of electricity classes 
itself with the other intensity-factors only by its forcefulness, 
with its gravitational tendency to fall ; for electrical action always 
takes place away from the locality or condition of higher voltage 
toward that of lower voltage. 

In chemical energy the intensity-factor is wanting, as yet. 
The only definite accomplishment in the line of identification is 
that of the direction of spontaneous chemical action as always 
coincident with increase in entropy. In this, chemical action is 
plainly in parallel with kinetic mechanical energy. But the bare 
existence of a duality of factors is as apparent in chemical 
energy as in the other forms. The mass factor is stated above 
as merely mass, whereas, to accord with mechanical energy, it 
should be the square of the mass. But here it must be remem- 
bered that chemical measurements concern themselves only with 
varying numbers of tiny mass-systems, each called a molecule, 
embodying energy in some particular chemical form of arrange- 
ment. Apparently the mass, energy and extent of mass-pairing 
of each molecule are the same ; the quantity-measurements have 
to do only with the number of molecules. In this case the 
quantity-factor must vary, not as mass-squared, but as mass. 

In thermal energy the two factors have already been suffi- 
ciently discussed. 

With the other forms of energy the identification of the two 
factors is still more obscure. Still, so far as our knowledge of 
these other energy-forms goes, it falls in line with the general 
concepts which were based upon mechanical energy in the earlier 
chapters. Thus, we know comparatively little about the mechan- 
ism of the sound-wave in air; yet we know that these waves 
produce pressure, due to the arrest of moving particles of mass. 
In the case of the radiant energy of light, also, the pressure 
developed by reflection has been observed by several independent 
observers. 

Even in the field of the most recently developed and least 
known of all the sciences, that of radioactivity, what little we 
know falls into line with these same fundamental concepts of 
energetic action. In radioactivity there are plainly two variables, 
the intensity of radiation and the mass. The extreme degree of 



216 ENERGY 

concentration itself, of energy within a given mass, is made 
more comprehensible by our earlier conclusions, drawn from 
mechanical energy, as to the unlimited ability of mass to em- 
body energy. 

Again, the intensity of radioactivity exhibits the most obvious 
downward tendency. One of the first tasks in identifying each 
newly discovered radioactive substance is to determine the time- 
rate at which the intensity drops. No such substance has been 
discovered in which the rate of radiation increased with time. 

Again, the intensity of all forms of radioaction decreases, 
with time, by what is known as the half-rate law. That is to say, 
its intensity decreases by one-half in equal portions of time. If 
the rate be called unity at any instant, and the period of time 
be observed until that rate shall have fallen to one-half, then at 
the end of the second equal period of time the rate will have 
fallen to one-quarter, at the end of the third period to one- 
eighth, and so on. If these rates should be plotted upon rec- 
tangular coordinates, with time for the other axis, there would 
result a curve of the form of an equilateral hyperbola, asymp- 
totic to both axes. In other words, the rate of radiation could 
never fall to zero, no matter how long it should continue to 
radiate; and, going backwards previously to the time of first 
observation, it may be said that the radiation must have begun 
within some fairly definite recent period, before which no radia- 
tion could have occurred without being infinite in its rate. In 
this the variation of radioactive intensity with time is quite 
similar to the form of variation, in hyperbolic function, of every 
other energetic function which has yet been examined in this 
series of papers. 

Again, many of the manifestations of radioactivity are based 
apparently upon the linear motion of very small particles of 
mass. As the velocity of these particles approaches that of light, 
186,000 miles per second, their mass apparently increases very 
rapidly. This is explained as due to the mass of the ether which 
must be displaced in their passage — just as the inertia of a ship 
is really that of the hull moving forward plus that of the water 
moving astern to make good its displacement. This explanation 
tacitly admits the massiveness of the ether. But even if this 
question be not entered, here is another energetic function — that 
between the apparent or effective mass of the moving particle 



TRANSFORMATIONS AND CONSERVATIONS 217 

and the difference between its velocity and that of Hght — which 
is hyperboHc in form. For apparently, if the velocity of light 
could ever be attained by these particles, their mass would have 
become infinite ; while for velocities far below that their apparent 
mass is very small. 

Again, the radioactive substances tend to degrade, with time, 
into substances more stable chemically. Both lead and copper 
are thought to have been produced from radium in this way. 
This variation of chemical mass-pairing and intensity, within a 
chemical mass which remains unchanged in the aggregate, is an 
apparent parallel with the way in which thermal mass-pairing, or 
entropy, varies within a mechanical aggregation of mass which 
itself remains unchanged. 

Energy-transformation. In all of these dual forms of 
energy, with factors varying in generally hyperbolic function 
asymptotic to two limiting zero-axes, the variation of each factor 
occurs in counterbalance against the other in stable equilibrium. 
Given sufficiently favorable conditions, this smooth fluctuation in 
stable equilibrium may cover an unlimited range. There is no 
known limit to the upward expansion of the intensity-factor, or 
the outward extension of the extensity-factor, except external 
conditions. 

These, in every natural case, dictate a certain limit to the 
exaggeration of either energy-factor beyond a certain point. 
The hyperbolic energy-function here becomes discontinuous. 
The energetic equilibrium, previously stable, becomes unstable. 
Energy-transformation sets in. The appearance of the energy 
to human senses is abruptly altered. The gravitational law pre- 
viously prevailing becomes invalid. It is only with especial care 
that the continuous force of even the laws of conservation — of 
mass, energy and intensity — may be discerned. 

In mechanical energy these limits of stable equilibrium were 
fully discussed, and their results, in the form of either collision 
or dissociation, were identified. In thermal energy these same 
limits are visible, but their exact nature must be inferred rather 
than observed. Of these the most familiar are the temperature- 
limits at which occur fusion, vaporization and chemical dissocia- 
tion. But these are wholly, or largely, internal in their equilib- 
rium. Fusion is almost independent of pressure ; vaporization 
is balanced against mechanical pressure; dissociation is balanced 



218 ENERGY 

against chemical pressure. In addition, thermal intensity is in 
equilibrium with mechanical pressure, determining whether work 
is to be performed by expansion or not. It is also in equilibrium, 
in the thermopile, with electrical intensity and resistance, deter- 
mining whether heat shall be altered into electrical energy or not. 
In solutions, temperature is in equilibrium with the proportionate 
presence of dissolved and undissolved substance. In ignitable 
explosives temperature is in equilibrium with the rigidity or 
strength of the chemical structure of the atom, so that when 
more than a certain amount of thermal energy is introduced, per 
unit of mass, the instability of equilibrium appears most spectacu- 
larly, in violence of explosion. In the detonating explosives is 
exhibited a similar equilibrium between mechanical intensity and 
chemical resistance. Dynamite may be heated to any degree 
whatever, so that it ignites and burns ; yet its chemical structure 
is stable (except for combustive action) in the face of this 
thermal intensity. But intensity of mechanical shock it cannot 
withstand, beyond a limited degree. 

Indeed, the familiar phenomenon of ignition is one of the 
best illustrations of energetic equilibrium. Up to the ignition- 
point a combustible substance absorbs heat in the same way as 
ice does — with gradual change in entropy and temperature, in 
stable equilibrium. But as the ignition-temperature is reached 
the chemical equilibrium becomes unstable, with a resultant 
transformation even more striking than when ice melts. 

But, since it has been said that all these illustrations of the 
more obscure energy-forms are explicable in terms of mechanical 
energy, the writer searched long for an instance of mechanical 
energy-transformation occurring, as the result of too great con- 
centration and intensity of energy, in some more familiar mechan- 
ical fashion than celestial collisions. It was finally found, one 
summer's day, while watching a steamboat describe a long circle 
across a still, deep harbor, under momentum only. The water 
was glassy calm. The bow-waves formed themselves, on the 
side of the boat toward the center of curvature, into a wide uni- 
form arc, which moved smoothly and noiselessly across the 
water's surface, narrowing concentrically as it came. The 
intensity of motion-energy in the water-waves was plainly being 
increased, by the concentric direction of the waves ; yet here was 
no disturbing question of external conditions, as when the wind 



TRANSFORMATIONS AND CONSERVATIONS 219 

drives a wave until it breaks, or a solid beach interferes with its 
progress and transforms its energy into heat. 

If our ideas as to instability of equilibrium resultant from 
undue intensity of energy were correct, something ought to 
happen here soon; and it did. Finally, almost simultaneously 
around the extensive arc, the waves broke noisily into foam, 
although the water was deep and the air was still, from sheer 
overconcentration of energy. A large portion of their energy 
suddenly became heat and sound. 

Stability in Energetic Transformations. Yet the insta- 
bility of equilibrium visible in energy-transformation is itself 
limited in scope. A pendulum with the bob held vertically above 
the point of support is in unstable equilibrium. Released, it will 
transform its store of energy abruptly, with positive acceleration. 
Yet the result of this action is to remove the phenomenon from 
the field of instability, into one where the equilibrium is stable. 

It is so with all energy-transformation. While energy- 
transformation is initiated only when the equilibrium is unstable, 
yet it occurs always in the direction of recovery of stability. 

In mechanical energy it was noted that too great intensity in 
the form of propinquity begets collision and energy-transforma- 
tion; but that the result of this is a form of energy, heat, in 
which occurs no collision. Too great an intensity of velocity, on 
the other hand, begets dissociation ; but the result of dissociation 
is to transfer the mass-portions into such propinquity to other, 
larger systems that they are trapped there and can no longer 
dissociate. 

Similarly with heat, unusual temperature begets energy- 
transformation in the form of work-performance ; but the first 
result of work-performance is to lower the temperature and 
stop the transformation. Unusual lack of temperature begets 
thermogic energy-transformation; and the first result of this is 
to develop entropy, volume and elasticity, so that the thermogy 
is retarded. 

The same is true of chemical energy, as familiarly visible in 
combustion. A combustible mixture of gases, if ignited, does 
not burn completely. Combustion is retarded by two things 
resultant from combustion : ( i ) temperature, causing disso- 
ciative tendencies which countervail the mutual attraction between 
fuel and oxygen; and (2) chemical pressure, due to the presence 



220 * ENERGY 

of a preponderating proportion of the stable chemical products 
of oxidation. Both resistances to further combustion rest upon 
forms of thermochemical equilibrium. 

Indeed, we are told that in such a mixture of gases there 
never exists a purity either of separation before combustion or 
of combination afterwards. That is to say, in every combustible 
mixture there exist before combustion some small portions of 
the products of combustion, in proportions determined by molec- 
ular equilibrium at low temperature. When the mixture is 
ignited most of the fuel and oxygen combine, but not all. A 
portion still remains in dissociation, held apart by the new condi- 
tion of thermal, chemical and mechanical molecular equilibrium, 
determined by the new temperature and pressure prevailing. For 
each different temperature and pressure there is a different pro- 
portion of fuel or oxygen still uncombined. Often in the arts, 
as in gas-enginery, this proportion is sufficient to be of economic 
importance. More often it is so small as to be more of a chem- 
ical curiosity. But always it is there. Apparently, no exaggera- 
tion of condition will either get all the fuel and oxygen apart, or 
induce them wholly to combine. 

In every phase of natural action this universal equilibrium is 
to be traced. The hot summer-day begets transformation of 
heat into electricity, breeds a thunder-storm and furnishes a most 
spectacular display of energy-transformation in unstable equilib- 
rium. But the thunder-storm ''cools the air" and recovers the 
weather's equilibrium ; and no more storms occur until heat again 
accumulates unusually. 

An electric current finds opportunity to enter a motor, finds 
there conditions favorable for transformation into motion — that 
is, an unusual intensity of field, coupled with unusual quantity 
of current across it. Motion ensues. But the immediate effect 
of the motion is to beget a counter-electromotive force, which 
so reduces the current that the transformation is reduced from a 
condition of unstable to one of stable equilibrium. 

A stream of water, flowing across a meadow, washes easily 
the soil from the banks and carries it with it. But the washing 
is done in unstable equilibrium. On whichever side it occurs, the 
water is thrown in that direction, by centrifugal force, and exag- 
gerates the departure from a straight line of flow. But, as these 
departures on either hand become extreme, the length of the 



TRANSFORMATIONS AND CONSERVATIONS 221 

water-course becomes exaggerated thereby. The hydrauHc 
"slope" of the stream becomes lessened, and its velocity of flow 
too low to carry longer an appreciable mass of suspended earth. 
So the stream adopts finally a series of S-shaped meanderings — 
in which the disposition to pick up soil is stably balanced against 
the disposition to drop it — as the form of water-course of per- 
manent equilibrium. 

In all departments of nature its every and most diverse 
aspect must be understood, first of all, as being the natural and 
inevitable result of preceding causes, acting always in stable 
equilibrium. The forms of not only earth, sea and solar system, 
but also those of vegetable and animal life, can be nothing else 
than the fruit of energetic evolution,' reacting with former self 
and present environment in an eternal stability of equilibrium. 
The known forms have survived because they are those embody- 
ing stability of equilibrium. 

In every field of activity known to man, in the energetics of 
moons, molecules and men themselves — in individual human life, 
in economics, in politics, in war and peace — the continued preva- 
lence of stable equilibrium and apparent quiet begets an accumu- 
lation of intensity which periodically surpasses the critical limit 
and begets instability. Spectacular transformation of energy 
ensues. But the instability is always temporary ; the transforma- 
tion always occurs in the direction of recovery of stability. 
Everything works in the direction of its own demise and the 
birth of a new regime. Natural phenomena never progress 
smoothly and continuously. In nature as in human history, in 
molecular as in military affairs, in the celestial chariots of 
Phoebus and Aurora as in a modern automobile, matters get 
ahead by a series of explosions, followed by relaxations into 
lassitude. 

It is the explosions, not the intervening periods of recupera- 
tion, which we perceive and by which we characterize the energy- 
form. To only one out of a hundred does "the French nation," 
for instance, mean aught or more than a Reign of Terror, a 
Waterloo and a Commune. The continuous daily life of the 
French people counts for nothing. Yet it is this continuous daily 
life which accumulates the energy become spectacular in the 
revolutions. 

It is thus that we must abandon hope of securing satisfactory 



222 ENERGY 

names or attributes for any one form of energy. It is only 
transformations of energy which appeal to us. We know nothing 
about heat as heat. It is only as it enters or leaves its cryptic 
ant-hill that we see it. When it transforms itself into nervous 
shock in our bodies, or into volume, pressure or electric current 
in our thermometers, or into work in our engines, or into light 
in our lamps, we say : "Lo, here is heat !" But in none of these 
cases is it the heat itself which we perceive. 

The Fundament of the Energetic Universe. It has been 
a constant care, in the preceding chapters, to dislodge the pre- 
vailing concept of energy as a flat-footed, static thing, resting 
upon an absolute zero of something as a supporting base, and 
rising therefrom in a simple, additive way. For this idea of 
energy — of all energy-forms, as well as for mechanical energy — a 
concept radically different, in two respects, is necessary. First, 
all energy-quantities vary on either side of a mean energetic 
value, which mean condition is itself unsupported. Secondly, the 
path of motion, in kinetics, as one of these variables, ranges thus, 
in eccentricity of conic-section orbit, on either side of the 
parabola, as the mean energetic path. The parabola is the funda- 
mental orbit, the sole natural geometric base for all things, the 
orbit of unit eccentricity, embodying equal quantities of radial 
and tangential motion. The straight line has no place in ener- 
getics. 

This needed metamorphosis of our ideas is so great as to 
appear impossible. Yet a quite similar transfiguration had to 
be, and was, accomplished in another science — astronomical 
kinematics — as much as three centuries ago. In the days when 
Vasco di Gama, Christopher Columbus, Magellan and Drake 
were opening the far seas to European commerce, and revolu- 
tionizing the world's ideas as to the nature and extent of its 
own civilization, what was the astronomical concept upon which 
rested their aids to navigation? The pre-Pythagorean or post- 
Ptolemaic, of a flat earth which served as a fundament for the 
heavens, relatively to which zero-plane of reference the sun 
"rose" and "set." By the time of Columbus the idea of the flat 
earth had given way before his own genius ; but the earth still 
remained as the center of interstellar space. And with the mass 
of people the idea of the flat earth continued tenaciously. It did 
not disappear from all of our Protestant church-creeds until 



TRANSFORMATIONS AND CONSERVATIONS 223 

within the last half-century. So prominent and able a man as 
President Kruger, of the Transvaal, still held to the ''simple" 
faith in the flat earth, the supported skies and the moving sun, as 
the twentieth century dawned — although the intricacies and 
inconsistencies into which it leads are beyond bare statement here. 

Columbus never met, in all his stormy voyages in tiny, top- 
heavy craft, natural obstacles to progress so great as he every- 
where encountered, in public opinion, in the prevalence of these 
crude ideas as to the astronomical nature of the universe. For 
generations navigation suffered untold loss because of public 
bigotry in refusing to countenance true astronomy. Even a 
century later than Columbus the Gallilean and Copernican philoso- 
phies almost carried their advocates to the stake ; and although 
the few then began to see, the rest followed slowly. The entire 
present wealth of these United States would not make good the 
losses to commerce and civilization which have been involved in 
the slow reluctance of mankind to abandon its reliance upon a 
rigid, tangible support for the heavenly bodies from an absolute 
base, in favor of a faith in intangible "action at a distance," and 
that too about an unsupported center, as a sufficient explanation 
of celestial mechanics. 

Yet this early astronomy was no more crude — in comparison 
with the Gallilean-Copernican concept of a central sun, itself 
moving, unsupported, through space, with the earth and planets 
revolving about it — than is the ''absolute zero," "up-and-down," 
rectilinear concept of energy^ as an attribute of homogeneous, 
indivisible, ultimate matter, which holds sway to-day, when com- 
pared with the truth. Many of the statements now taught to 
our youth as fundamental principles of mechanical science are 
the exact reverse of the truth. 

Yet the time of reform is now upon us. Some vital change 
looms imminent. Energy is now as important a topic as navigation 
was then. The great industrial and monetary interests are now 
linked with the use of natural energy in manufacture, and with 
the manufacture and sale of energy itself, as they were then 
with transoceanic discovery and commerce. Just as navigation 
was then the prime factor in gigantic transformations in human 
thought and political institutions, so is discovery in the field of 
energetics now the guiding cause in enormous recent and immi- 
nent changes in pubHc opinion and democratic institutions. It 



224 ENERGY 

was steam-transportation, the cotton-gin and the telegraph which 
fifty years ago made of slavery — an institution which had existed 
beneficently to man since the dawn of History — an anachronism 
so inefficient and disturbing that its abolition was forced upon 
the nation, upon civilization, at whatever cost in men and money. 
The similar or greater changes in the form of our social organiza- 
tion which now promise to be forced upon us, as the inevitable 
result of the more recent discoveries of the telephone and trolley, 
the gas-engine and the steam-turbine, hydro-electric transmission- 
systems for light and power, wireless telegraphy and rural free 
delivery, are yet to be measured out in nature's laboratory. 
For their thorough comprehension and their safe guidance it is 
imperative that the run of practical men of afifairs should possess 
accurate concepts of the internal energetic action and possibilities 
of large and intricately organized masses, whether of molecules 
or of men. But before we may hope to step into such a true 
comprehension of the energetic universe, purified from its present 
chaotic mixture of inconsistency with complexity, we must alter 
our point of view from its present post-Ptolemaic to a more 
Copernican position. We must get off the surface of the earth 
and rise above every-day human standards, before we may grasp 
the significance and the majesty of that every-day phenomenon : 
energy-transformation. Universal law holds true here, as else- 
where ; but we, with our little factories and heat-engines, are not 
the fundament, nor even the center, of the universe. 

It may be true, as says the writer on astronomy in the 
Encyclopedia Britannica, that the Copernican feat of removing 
the center of the celestial system from the earth to the sun, with 
its immediate unfolding of the complex mystery of the planetary 
system into rational simplicity, accomplished no perceptible 
advance in the science. It may be true that the future of physics 
lies solely ''in the sixth decimal place." The writer does not 
believe either statement. The sort of astronomy which knows 
nothing outside of the sixth decimal place possibly was not 
advanced by Copernicus. But the sort of astronomy which could 
never have been revealed by sixteen decimal places, applied to 
the old ways — the sort of astronomy which fires men's minds 
with new ideals and devotions, which tears inside out old world- 
systems of bigoted faith and cruel superstition — this sort could 
never have lived without Copernicus and Gallileo. 



TRANSFORMATIONS AND CONSERVATIONS 225 

But even the coldest and most mathematical science pro- 
gresses thus. The re-definition of terms, the codification of laws 
and the projection of rational hypotheses are all as powerful aids 
to efficient observation as is the latter to the accurate growth of 
theory. And just at present — particularly in both the engineering 
and the economic fields — the empirical side is unquestionably 
overpulling on the whifile-tree; we possess far more data than 
we have yet properly digested. 

The Unity of All Energetic Action. There can be little 
question that the present broad trend of scientific progress is 
along the lines of an accumulation of complexity of detailed data, 
but with a simultaneous precipitation therefrom of an increas- 
ingly simple, consistent and unified set of underlying principles. 
This is true not only in pure science, but also in engineering and 
the other applied sciences — and, above all, in sociology, the most 
important of all sciences to human happiness. The problem has 
nowhere been better stated than by Sir William Ramsay, in his 
1904 address before the St. Louis meeting of the International 
Congress of Arts and Sciences, in discussing the imminent prob- 
lems of chemical science : 

'T have already, in an address to the German Association at 
Cassel, given an outline of the grand problem which awaits 
solution. It can be stated shortly, then : While the factors of 
kinetic and gravitational energy, velocity and momentum, on the 
one hand, and force and distance on the other, are simply related 
to each other, the capacity factors of other sorts of energy — • 
surface, in the case of surface-energy; volume, in the case of 
volume energy ; entropy, for heat ; electric capacity, when electric 
charges are being conveyed by means of ions ; atomic weight, 
when chemical energy is being gained or lost — all these are 
simply connected with the fundamental chemical capacity, atomic 
weight, or mass. The periodic arrangement is an attempt to 
bring the two sets of capacity-factors into a simple relation to 
each other ; and while the attempt is in so far a success, inasmuch 
as it is evident that some law is indicated, the divergences are 
such as to show that finality has not been attained. The central 
problem in inorganic chemistry is to answer the question, Why 
this incomplete concordance?" 

But is it a fact, as Sir William states, that the factors of 
mechanical energy are so simply related? Is it not true that 



226 ENERGY 

other sciences are obscure chiefly because our mechanical con- 
cepts are confused, vague and often inconsistent? Is it not 
likely that, when we have swept our eyes clear of cobwebs in 
regarding our more familiar forms of energy, the more obscure 
ones may stand out in much better definition? At any rate, to 
do anything, however imperfect, toward the improvement of our 
scientific basis for this broader aspect of all the natural sciences, 
as mere departments of a single, consistent whole, is the highest 
aim to which human thought may now aspire. 

Indeed, this is the basic object of all true education — as dis- 
tinguished from mere training — to open the eyes to the invisible, 
to broaden the narrowness of view of ignorance. For this there 
is needed only an early inculcation of the unity of all nature. 
We do not hesitate to place early in the high-school course the 
doctrine of the Conservation of Matter, in spite of the infinite 
variety of form in which matter appears. We regard the doc- 
trine of the Conservation of Energy, throughout similar diversity 
of form, as the core of our college-taught science. Why should 
not the parallel doctrines of the Conservation of Intensity, of 
the Duality of Energy-factors, and of the unity of all extent- 
factors with mass-subdivision, be taught as equally basic concepts ? 

To many writers, too, the assumption seems to come nat- 
urally that the different localities and scientific departments of 
universal action are quite independent of each other, or even 
discordant. The fact of unity, interdependence and identity 
seems to call for some rigid proof, before it can be accepted. 
They seem to forget that basic principles are always axiomatic. 
They seem to forget that a ''proof" is nothing more than the 
dependence of a conclusion upon its premises, and not possibly 
of greater import than those premises. But to the writer the 
identity of all sorts of natural action lies in the axiomatic 
premises. It is their discordance which must be proven. Since 
the discovery of universal gravitation and the speed of trans- 
mission of light the universe has been unified over distances 
hopelessly beyond human comprehension, by bonds measurable, 
as to time, in terms of human life and action. Since the dis- 
covery of the mutual interchangeability of light with heat, 
motion, animal and vegetable life and inanimate electricity, since 
the invention of the spectroscope and the bolometer, the unity 
and identity of natural action in the most remote abysses of 



TRANSFORMATIONS AND CONSERVATIONS 227 

interstellar space with the most familiar of every-day happenings 
here upon the surface of the earth have become, as was said, 
axiomatic. Their underlying principles are not merely similar; 
they are palpably identical. The burden which lies upon us is 
not that of proving that they are identical. It is, rather, to define 
in detail what are their differences. 

Nor does this idea of unity mean that all forms of energy are 
but allotropic forms of one basic form, whether that be electrical 
or mechanical or chemical. It means that each is a different 
outward aspect of a single hidden inner nature, which latter we 
may never hope to comprehend. To the writer, mechanical is 
the most familiar form of energy; therefore he naturally refers 
all other extents of energy to mechanical pairs of mass, and 
all other intensities to visible space and motion. Yet he does 
not know what either mass, or space, or motion really is, and 
has no expectation or desire that any one will ever know. 
Similarly, to Professor J. J. Thomson, for instance, electrical 
is the most familiar form of energy; so he naturally refers all 
other extents of energy to electrical charge as a base. Yet he 
makes no pretence, I believe, that the true inner nature of the 
electrical charge will ever be known, however minutely we may 
dissect it further in the future. To Professor Ramsay, again, 
chemical energy is the natural base. Yet here again is no better 
hope of ultimate comprehension. Mere reduction into terms of 
something else is all that science may ever attempt. The unity, 
but not the ultimacy, of nature is the lesson of science. 

"Energy," then, is a dual circular chain of links. Each 
"form" of energy constitutes a link in the circle. As we walk 
about this circle we may regard the different links lying nearest 
us, the chemical or thermal or mechanical as may be. From 
these combined impressions we judge the inward nature as best 
we may; just as we know an actor only after seeing him in 
many parts, under diverse make-ups. But none may say that 
any one of these make-ups is the actor himself. 

In this great movement of human thought and action, due to 
civilization's current change of front from its earlier material- 
spiritual basis to its present ultra-energetic aspect, it is as 
natural that the engineer should forge to the front, as the first 
to understand and to do, as it was in earlier times for soldiers, 



228 ENERGY 

navigators and lawyers to be the leaders of men. But, if he is 
to rise to his opportunity in the new century, the engineer must 
be more broadly equipped. He must understand not only 
machines and individual men, but vast masses of men — not 
hundreds or thousands of them, but miUions and tens of millions 
of them. As his first start toward equipment for his public and 
private duty he must grasp the great, fundamental principles of 
all energetic action. He must understand, as well as memorize, 
these three basic laws of all natural action, viz : 

First: All energetic action, whether classed as celestial, 
mechanical, thermal, chemical, electrical, biological or sociological, 
operates under the same general principles in action. For in all 
these diverse forms, in so far as anything exact may be said 
about their structure, energy consists in the subdivision and 
organisation, into specified relationships of motion and arrange- 
ment, of a mass of material. Yet this material, of itself, pos- 
sesses none of the characteristics peculiar to the zvhole. The 
nature of a celestial system is not determined by the peculiarities 
of its planets, but by the peculiarities of their orbital relation- 
ships. That of a machine does not depend upon those of its 
component parts (so long as they come up to certain minimum 
requirements), but upon the way in which the engineer has put 
them together. The features of a chemical compound have 
nothing to do with those of its component elements. Gases can 
be combined to produce a solid, and solids to produce a gas, and 
vastly greater contrasts between raw material and result con- 
stantly appear in the chemical laboratory. Our deadliest poisons 
and best foods are both but carbon, hydrogen, nitrogen and oxy- 
gen, differently arranged. All are explained as being different 
relationships, within the molecule, of elementary atoms which 
are alike for all known chemical substances. 

In the most varied physical aspects of electrical action it is 
the same. No one has gone further than Prof. J. J. Thomson 
in the reduction of all phenomena to mere variations in relation- 
ship between elementary components possessing only elementary 
characteristics ; yet he works chiefly with electrical concepts. 
Similarly, the most intricate variety of vegetable forms of life 
is shown by botanists to be but a variety of arrangements of a 
substantially uniform vegetable cell, which is specialized about as 
much to develop root, stalk, leaf or flower in any one plant as it 



TRANSFORMATIONS AND CONSERVATIONS 229 

is to embody the vast differences between one plant and another. 
And the same is true, to skip briefly to the other extreme of 
natural action, of the energetic action of men. One method of 
organization will make of an army a panic stricken mob ; another 
constitutes it an invincible foe. One plan of organization within 
a factory leads to chronic bankruptcy ; another to opulent profits. 
Anthropologists tell us that the peasants of modern France, which 
leads the world in science and art, are the exact copies, in cranial 
development, of their ancestors of eighty thousand years ago. 
But eighty thousand years ago ideas of political and economic, 
organization were exceedingly crude. 

The one necessary lesson for the clarifying of future progress 
is that each form of energy is defined, fitly for scientific dis- 
cussion, only when we consider activities between its component 
units, carefully excluding all action which may occur zvithin any 
"unit." Otherwise is confusion and no progress. The "unit" 
may be an electron, or a molecule, or a solar system, or a pro- 
toplasmic cell (in biological energy), or a man, as in sociological 
energy. Or, in the case of that international energetic action 
and reaction which has arisen with the steamship, the cable and 
the wireless, the unit of mass may be an entire nation. The 
true energy, existing between the units, may trade energy with 
its component units, or with external systems, it is true; but 
unless we exclude these sources and destinations from the dis- 
cussion we are talking of two or three things at once. Confusion 
is inevitable. 

Nowhere is this need of definition and clarity greater than in 
discussing the sociological energy existent between (not within) 
individual men. When we assemble a hundred metallic parts 
into a machine we regard the relationships between the parts as a 
special form of energy — mechanical — and as a fit subject for a 
special science, mechanics. We leave all metallurgical questions 
lying within each piece to other books and men. Electricity, 
again, we define as a relationship between electrons, positive and 
negative. But when we get a few millions of electrons cemented 
into a number of molecules, we call the relationship between 
the molecules a new and distinct form of energy — chemical. 
When we get a billion molecules organized into a protoplasmic 
cell or so, we call this relationship between the different chemicals 
still another form of energy — biological. When we get a million 



230 ENERGY 

protoplasmic cells arranged in organic form, and the organs spe- 
cialized and federated into a sentient, reproductive animal, we 
regard the relationship between the cells and organs (which are 
all alike, yet all different) as still another distinctive form of 
energy — that of the human individual. 

But when we get a hundred million men and women organ- 
ized into special sexes, ages, trades and professions, and these 
federated and refederated into a modern State, we decline to 
admit that there arises therein a new and distinct form of life 
and energ}^ — the sociological — between individuals. We insist 
upon dragging into the question at every point the human nature 
within each unit — each different, it is true, yet averaging as like 
as any million molecules. We decline to see that human institu- 
tions may themselves have an organic life, growth, reproduction 
and death — as independent of the myriad of individual lives, 
growths, passions and deaths within them, as the history of our 
solar system as a whole is independent of the internal natures 
of its component parts ; which last are much more diverse than 
are dift'erent human natures. We fail to see that the written 
history of mankind records the growth, not of individual man — 
for evolutionary science declares this growth to have been com- 
pleted before the dawn of history — but of this institutional organ- 
ism of human relationships, an organism as distinct from indi- 
vidual man as the latter is distinct from his own component 
protoplasmic cells. That is why we fail, at present, of a con- 
sistent and satisfactory science of sociology: we have not yet 
taken it up as a department of universal energetics. 

With our civilization now approaching a feverish paradox of 
farce and tragedy, with stupendous rates of production and 
transportation of the means for life rising in rivalry with stupe- 
fying rates of poverty, suicide, insanity and crime; with our 
cost of living rising while labor-saving aids multiply; with our 
system of exchange left as religiously to the care of chaotic antag- 
onism of interests and duplication of effort as our systems of 
production have been subjected to the last refinement of coopera- 
tive organization — with all these phenomena becoming the char- 
acteristic ones of our world-civilization, the sociological doctors 
disagree, both as to diagnosis and remedy, more and more hope- 
lessly. It is high time that the said doctors were sent back to 
school, and there impregnated with a vigorous concept of the 



TRANSFORMATIONS AND CONSERVATIONS 231 

general principles of all energetic action. These apply most 
effectively to every other problem in a pretty wide and intricate 
universe. They will solve our sociological problems. They 
should be made the fundament of every college-course, whether 
aimed at pure science or at pedagogy, at engineering, medicine, 
law, the ministry, journalism or statecraft, as the ultimate goal. 

Secondly: All energetic action consists in a swing of one of 
the two great energetic factors — number of correlated parts, on 
the one hand, or intensity of relationship, on the other — on either 
side of 'a central, or mean energetic, condition. In no case may 
any of the factors ever reach zero ; none may ever reach infinity. 
And this swing occurs under the guidance and propulsion, as 
also against the resistance, of two great gravitational tendencies, 
one toward the consolidation and the other toward the disgrega- 
tion of the component material. 

These two gravitational tendencies are never directly opposed. 
Each is disposed laterally or transversely to the other. Within 
certain limits, each may act independently of the other. Between 
the two, therefore, may occur the greatest variety of lateral per- 
turbations of the general swing from one extreme to the other. 
The pendulum of energetic conditions is not confined to a single 
plane, but is capable of the utmost variety of cyclical gyrations — - 
never confining itself to any regular geometric path, and seem- 
ingly intricate in its motions beyond comprehension — yet always 
guided in an eternal stability of equilibrium. In whatever direction 
departure may be made, the departure itself begets the disposition 
to return. Whether the swing be confined within the critical limits, 
concerned merely In an exchange of one dimension of energy for 
the other, or whether it trespass beyond, begetting energv'- 
transformation, the equilibrium continues ever stable. There Is 
no activity in nature, inanimate or animate, which does not vary 
stably about a mean central condition, from which It never can be 
driven, by vagary of circumstance, more than a finite distance, 
or against less than a proportionally Increasing resistance which 
must eventually reverse the process into a return. 

Thirdly: This central or mean energetic condition, which 
neither requires nor is capable of any fixed support from any 
rigid base, but hangs in mid-space like the sun in the heavens, 
contains always an Immeasurable amount of latent and Invisible 
"tangential" energy, into which and out of which the perceptible 



232 ENERGY 

or "sensible" funds of radial energy pass in indefinite amount. 
There is no department of natural action which has proven so 
deceptive to the engineer as that of latent energy. The instances 
are legion. The most dramatic illustrations can be drawn, as 
usual, from social energetic systems, wherein the politicians, if 
not the statesmen, are continually deceived as to the energetic 
possibilities latent within a people, and as to when they are likely 
to burst forth. In 1776 Great Britain was nonplussed by the 
indomitable resistance of a few ragged American farmers. In 
1795, when France at home was a howling mob, unable to enforce 
order or supply bread on the streets of Paris, her armies scat- 
tered the combined forces of England, Prussia, Austria and 
Italy. In America again, in 1864, the South and the copperheads 
could not understand whence came the unending resistance of 
the North — though Gladstone's insight told him that the South, 
with England behind it, was "fighting the law of gravitation." 

To-day the same is true. The engineers are as blind as the 
politicians, in their failure to comprehend the enormous latent 
possibilities for productive energetic action which lie in the 
armies of workmen of which they must always be the officers. 
If only these armies be once organized for a single harmonious 
end, their energy, according to the laws of energetics, must 
increase more rapidly than the first power, and possibly as fast 
as the second power, of the numbers involved. That single end 
— in order to develop this best rate of increase — must be neither 
mere volume of material output nor accentuation of dividends, 
although to-day these are the sole aim of the engineer-adminis- 
trator. It must be, instead, the welfare of the consumer, for 
whose support alone exists the entire economic system. The 
present division of society, in public opinion, into several classes, 
such as laborer, employer, capitalist, etc., of which the consumer 
appears as merely one, must cease. In economic democracy, 
none of these possesses any rights whatever except as a con- 
sumer. In primitive nature the law of hunger drove the arm 
of toil inexorably. In intricate societies it must be the same. 
Yet in modern economics the law of supply and demand operates 
but remotely, obscurely and indirectly. Like a river under- 
ground, or one obstructed by dams and dikes and diverted into 
artificial channels, it flows and feeds. But that is all. It is not 
free and it does not control. 



TRANSFORMATIONS AND CONSERVATIONS 233 

It is just as natural that a growing population should be 
increasingly self-supporting as it is that a locomotive or steam- 
ship should be increasingly able to haul more coal than it burns. 
But if we should design our locomotives for the much more 
spectacular purpose of maintaining a pyrotechnic display of 
sparks by night and a salvo of steam-fountains by day, rather 
than for coal-hauling, we might easily find that they could not 
haul coal enough even to supply their own consumption. The 
more such locomotives we built the poorer we should be. 

Yet our present method of constructing economic systems — • 
or rather, of tolerating unchanged those which we have inherited 
from an ignorant past — is quite as this. The real and sole reason 
for the existence of an economic system at all — the transfer of 
food from the soil to the mouth — has been almost totally for- 
gotten. All now centers upon considerations of immediate profit 
from intermediate exchange, multiphed for the purpose. Our 
economic system operates primarily for the purposes of profitable 
antagonism, empty display and concealed gains — with huge inci- 
dental waste. The consumer is supposed to have all proper control 
over those activities, which his money alone hires into being, 
when, at the bargain-counter, he chooses between two or more 
parallel lines of those activities ; which lines are generally as like 
as two peas. The fact that he alone pays all the bills, for raw 
material, labor, superintendence, finance, dividends, profits, and 
finally the cost of persuading himself to buy what he actively 
desires or urgently needs, and that therefore his right to control 
goes as far as does his dollar, has been quite forgotten. 

Instead, intermediate means are confused, as guides in organ- 
ization, with these sole ends — the sustenance and profit of the 
consumer. The costly strife for private profit at intermediate 
points, the purposeless paying of dividends for the purpose of 
enticing into reinvestment a remnant of those same dividends, 
under the guise of timid, though very "willin' " capital, and for 
the support of a pyrotechnic spectacle of luxury which forms no 
essential part of production and distribution, has overshadowed 
the sustenance of the real life of the community as the sole object 
of all business. Incidentally thereto has arisen, with the increasing 
complexity of invention and the arts, a rapidly increasing intri- 
cacy and intensity of confusion between all industrial organiza-, 
tions, which no business-man would tolerate for a moment 



234 ENERGY 

within any one of them which he controlled. With the com- 
niercial and technical press daily calling with greater vehemence 
for refinement of factory-organization, for better efficiency of 
result, the productive organization of the community as a whole 
is daily presenting a more hopeless chaos of cross-purposes, 
antagonism of interest and duplication of effort, resulting in a 
rapidly decreasing efficiency of result — the feeding of the con- 
sumer. With labor-saving and luxury-creating invention advanc- 
ing at a rate never before known in history, the cost, difficulty, 
uncertainty and dissatisfaction of living are daily upon the 
increase.* 

It is the organization, therefore, of all industrial enterprises 



*Should the reader be interested in following more in detail this grow- 
ing inefficiency of our system of industry and exchange, he will find the 
same analyzed and displayed, for the half-century of American progress 
from 1850 to 1900, in the writer's "Cost of Competition." He will find 
there the proofs that, whereas in 1850 the efficiency of organization — quite 
aside from any question of efficiency of individual effort — was such that 
seventy per cent, of the effort exerted was transformed into useful result, 
while thirty per cent, was dissipated in commercial impact and friction, 
over questions of price and ownership which are of no interest whatever 
to the consumer, so, in contrast, in 1900 these figures had become almost 
exactly reversed. Of the effort now being expended within our commercial 
system less than one-third results in useful product; and that small frac- 
tion suffices to produce all which we now consume. Of this same effort 
actually expended fully seventy per cent, is being currently lost, in impact 
and friction due to sheer lack of intelligent organization between factory 
and factory. 

The daily progress of our times, as revealed in the weekly reviews, can- 
not be understood except from these facts as a basis : that with the issue 
of every new patent, with the landing of each new immigrant, life grows 
more complex. The demand for extension of organization grows more 
urgent. It is neither the poverty nor the criminality of the immigrant 
that is the trouble. Both prove, upon rigid investigation, to be imaginary. 
The immigrant is a raw material of the greatest potential value. But, 
dumped upon a land wherein economic organization is proceeding at a rate 
far below that required by natural law — far below all other rates of 
progress — its effect is that of a ton of coal dumped on a furnace-fire need- 
ing a hundred-weip-ht. Valuable as are both invention and immigration, 
it is their combination which is now forcing the country into economic 
instability. We are much further from assimilating properly, by industrial 
combination, the current influx of invention than we are from that of immi- 
gration. Beneficial as is the industrial combination of tangible oroper- 
ties into greater unitv — for it is the major source of our wonderful eco- 
nomic progress in the recent past — it is now proceeding at a rate far below 
that requisite for the control of social intensities below the critical point, 
for the preservation of stability of equilibrium and for the prevention of 
explosive energy-transformations as far-reachine and destructive as those 
involved in the abolition of slaverv. The only solution lies in the far more 
rapid unification of all industrial properties — at a rate nearly proportional 
to the square of the population. 



TRANSFORMATIONS AND CONSERVATIONS 235 

into a single whole, along exactly the same lines as those now 
enforced by all business-men in the organization of individual 
men within these enterprises, which alone can develop within the 
community its quantity-factor of social energy into commen- 
suracy with its growing needs, which alone can expand its pro- 
ductive capacity more than proportionally to the first power of 
its population. It is the engineers of the community, more than 
any other one class, who must perform this task. It is they, 
above all others, who are equipped for understanding what they 
are about as they do this thing. 

If this task be not undertaken the critical limits of accumu- 
lated intensity will soon be passed. Intelligence will then have 
become impotent. Forces will have been released which must 
then have their brutal sway uncontrollably, until stability be 
regained through exhaustion. Economic equilibrium, already 
wavering, will have become grossly unstable. Explosion must 
ensue. It is not that labor will strike successfully against 
capital. There were as many chances for that three centuries 
ago as now. It is that the hundred million consumers will 
strike against the absurd strife and confusion now prevailing, 
not only between labor and capital, but between capital and 
capital, leading to such universal impact and friction that ineffi- 
ciency of result is growing upon us apace. They will rise and 
overthrow in its entirety a system which, in the twentieth cen- 
tury, can still find no better foundation and guide than universal 
antagonism of interest between man and man, between enterprise 
and enterprise — an antagonism to-day artificially stimulated to 
the last degree, in a vain endeavor to rouse it to a task for 
which it is inherently and inevitably impotent. No imaginable 
expansion of intensity of economic energy can ever meet a need 
for its greater extensity. Its only effect can be to exaggerate, 
as it delays, the vigor of the inevitable reaction. 

We shall be surprised, in looking back upon this crisis when 
it is passed, to see how largely its incurrence has been due to 
the fact that the business-men, factory-superintendents and 
mechanical engineers of a mechanical people do not understand 
the true nature of mechanical and allied energies. 

Conservations. Throughout all this wonderful intricacy of 
energy-transformation, between work, heat, chemical and elec- 
trical action, light, radioactivity, vegetation, animal life, the 



236 ENERGY 

activities of the body politic and economic, and the latent strength 
of that vast whirl of international human solidarity — the accumu- 
lation which constitutes, in the eyes of mankind, the highest aim 
of all these combined — throughout this whole, for all time, run 
the three great principles of eternal conservation, the first scien- 
tific statements of immortality. All that we can see or know as a 
Thing, throughout all this limitless intricacy of things, proves, 
upon examination, to be a mere temporary form. It is a form 
of relationship, between component portions none of which pos- 
sesses the attributes or abilities of the Thing itself. These attri- 
butes and abilities are the property of the forrn of relationship 
onlv. As this form is created, either from formless dissociate 
dust by its congregation into organized, interacting propinquity, 
or from senseless solidity by its comminution and disgregation 
into energetic sensitiveness, these attributes and abilities come 
into existence. As the form of relationship changes, so do the 
attributes and abilities. As the form of relationship melts again 
into formless dissociation, or solidifies into passive stolidity, the 
attributes and abilities disappear. 

Creation, birth, life and death are of form only. The one 
life of the universe continues unceasingly and unvaryingly. It 
is the reality alone, not the form, of the universe which is 
eternal. It is that, too, which is imperceptible; which possesses 
neither attributes, character nor individuality. It is not alone 
that the human senses take no cognisance of aught but mere 
form. They take cognisance only of change in form. If it 
were not for the birth, life and death of many millions of ether- 
forms before the eye each second, we should see no light. Were 
it not for ceaseless alteration of air-pressure upon our ear-drums 
we should live in blank silence. Were it not for ceaseless chem- 
ical metabolism of carbon, hydrogen, oxygen and nitrogen — 
themselves unchanging — in vegetable life, we should live in a 
desert; though sun, moon and earth still circled, we should have 
no seasons; winter would mean a cold bare rock and summer a 
hot one, equally bare. Were it not for the ceaseless birth, change 
and death of countless cells comprising our own bodies, so that- 
we possess none of the flesh which we inhabited a few months 
ago, we should know no individual life and growth. Were it 
not for the ceaseless procession of new-born babies into the 
world, of children shooting up into the bloom of adolescence, of 



TRANSFORMATIONS AND CONSERVATIONS 237 

active lives grown seamed and scarred and feeble from the buffets 
of fate, of old people laid lovingly away to rest — there could be 
no progress and history of the human race. 

What folly, then, to speak of inducing progress ! Progress 
occurs because it cannot help itself. Behind it is the energy of 
countless eons, non-creatable and indestructible. As well invite 
the earth to move more rapidly about the sun, as well invite the 
vine to grow more rapidly than its natural rate for soil and sun 
provided, as to attempt to invite or force — or quell or retard, for 
that matter — human progress. Not only the energy, but also 
the forward motion, of the race is indestructible and non- 
creatable. As mankind entire is but a bit of microscopic growth 
on the surface of a tiny mass-portion whirling in space, the last 
and most delicate fruit of ages of upward struggle on the part 
of trilobite and dinosaur, so are his energies but the latest form 
and conservation of the measureless energy-fund of the universe, 
whirled back and forth across abysmal space with inconceivable 
speed, but incapable of being lost or retarded, or increased or 
accelerated, by the slightest iota. 

The fundamental principles of energetic conservation upon 
which these conclusions rest are these — stated in terms of 
mechanical energy, as the simplest and most familiar form, but 
interpretable in terms of any and all known energy-forms. 

I. The FIRST Law^ of Energetics: the Conservation of 
MASS. Mass is quantity of matter. It exists eternally. It 
undergoes local and temporary aggregation or disgregation, 
ceaselessly ; but it is never destroyed nor created. 

II. The SECOND Law of Energetics: the Conserva- 
tion of ENERGY. Energy is the space-and-motion relationship 
between separate portions of mass. It exists eternally. It 
undergoes local and temporary accumulation, dissipation and 
transformation, ceaselessly ; but it is never destroyed nor created. 

III. The THIRD Law of Energetics: the Conservation 
of INTENSITY or AVAILABILITY of Energy. Intensity 
is the degree of spacial propinquity and of linear motion between 
the separate mass-portions. It exists eternally. It undergoes 
local and temporary concentration or diffusion, ceaselessly; but 
it is never destroyed nor created. 



238 ENERGY 

IV. The FOURTH Law of Energetics: the Conserva- 
tion of EXTENSITY of Energy. Extensity is the extent of 
relationship, or mass-pairing, between the interacting portions of 
mass. It is what embodies the intensity of energy, to give the 
latter a habitation and a name with which to do its work. It 
exists eternally. It undergoes local and temporary accentuation 
or disguise, through the aggregation or disgregation of mass, 
ceaselessly ; but its sum total is never created nor destroyed. 



THE END. 



INDEX 

Page 
Absolute zero 79 

Adiabatic 130 

Angle of Incidence 24, 35 

Apastron 22 

Basic Energetic Processes 129, 156, 163 

Carnot cycle 166 

Clausius 136 

Combative energy 55 

Conservations 208, 235 

Conservation of Energy 18, 137, 237 

of Mass 18, 237 

Contact 27, 114 

Copernican philosophy 223 

Critical energetic conditions 61, 66 

limits of intensity 66, 218 

Cycle 158 

Cycle-efRciency 169, 171 

Density 126 

Dimensions of energy 48, 78, 162, 214 

Dualism in energetics 48, 78, 162, 213 

Eccentricity 25, 34, 47 

Elasticity 90, 122, 143 

Energetics, Dualism in 213 

Elements of 17,30,46,74,76,78,163,237 

Energetic Action. Unity of all 225 

Cycle 158 

Equilibrium 84, 195, 217, 231 

Form. Interchangeability of 209, 211 

Gravitation 195 

Universe. Fundament of ^ • 222 

Energy- fund 32, 35 

Energy-transfer 1 12 

Energy-transformation 217 

Cause of 79, 87 

Equilibrium in 203 

Fundamental equation for 18 



INDEX — Continued 

Page 

Entropy 100, 131, 136, 142, 151, 156, 157 

Athletic 155 

Combative or military 55 

Entropy-temperature diagram 95, 100, 136, 190 

Equilibrium. Energetic 84, 203, 219 

Between interchangeable forms of energy 203,219 

Intramolecular , 152 

Thermal 182 

Extensity of energy 48, 51, 78, 162 

Extreme energetic conditions 61 

Factors of energy 48, 78, 162, 214 

First law of energetics 31, 237 

Fourth law of energetics 238 

Free motion 18, 30 

Fundament of the energetic universe 222 

Gravitation. Energetic 81, 195, 204 

Mechanical 13, 67, 196 

Thermal 105 

Heat 89, 144 

Heat-transfer 129, 134, 143 

Inelasticity 90 

Intensity of energy 48, 78, 162, 195 

Gravitation of 204 

Rejuvenation of 203 

Summation of 205 

Interchangeability of energetic form 211 

Irregular cycles 175, 178 

Isomorphic 98 

Isothermal 99 

Kepler's laws 29 

Kepler's and Newton's laws combined 30 

Kinetic energy 15 

Labority 135, 143, 149, 156 

Latent heat 99, 169 

Lovv^er critical intensity 66 

Mass, Conservation of 18, 237 

Mass-pairing 48, 51, 139, 154, 173 

Mean energetic condition 32, 79, 81, 82, 190, 222, 231 

distance 33 



INDEX — Continued 

Page 

Mechanical energy 9, 30, 78 

Basic processes of 163 

Mechanical universe 30 

Metamorphic 98 

Metathermal 98, 129 

Military energy 55 

Natural action 30 

Newton's laws 13, 28 

Newton's and Kepler's laws combined 30 

Parabola 25, 34, 76, 222 

Periastron 22 

Permanent energy 74, 93, 202 

Potential energy 14, 41 

Pressure. Mechanical concept of ... : 106, 145 

Primary and secondary energy- forms 175 

Propinquity 41 

Radial energy 35, 40, 143 

intensity 65 

Rejuvenation of energy 203 

Relativity 59, 228, 236 

Reversed cycles 175 

Reversibility 136, 172 

Second law of energetics 31, 237 

of thermodynamics 205 

Sensible heat 99 

Stability of energetic equilibrium 84, 203, 219 

Subpermanent and superpermanent energies 74 

Summation of energetic intensities 205 

Tangential energy 35, 41, 143 

Temperature 95, 141, 145, 151, 156 

Thermal conduction 129 

Diagram 95, 100, 190 

gravitation 195 

energetics. Basic processes of 129, 156 

equilibrium 182, 201 

Thermogy 134, 143, 148, 156 

Third law of energetics 237 

Transformation of energy 79, 87, 208 

Vibratory energies 18^ 

Volume. Mechanical concept of 106, 145 

Water-wheel cycle 160 

Wire-drawing 130 

Work-performance 129, 135, 143, 149, 156 

Zero. Absolute 79, 98 



f^ov so Md 



DEC , a i ^ ^ 



